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I will present aspects of a theorem, joint with Mihalis Dafermos, Gustav Holzegel and Igor Rodnianski, on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes.

March 29 – April 1, 2022: 10:00am – 12:00pm ET, each day

Location: Hybrid. CMSA main seminar room, G-10. Zoom link will be available.

Pieter Blue, University of Edinburgh, UK (virtual)

Title: Linear stability of the Kerr spacetime in the outgoing radiation gauge

Abstract: This talk will discuss a new gauge condition (i.e. coordinate condition) for the Einstein equation, the linearisation of the Einstein equation in this gauge, and the decay of solutions to the linearised Einstein equation around Kerr black holes in this gauge. The stability of the family of Kerr black holes under the evolution generated by the Einstein equation is a long-standing problem in mathematical relativity. In 1972, Teukolsky discovered equations governing certain components of the linearised curvature that are invariant under linearised gague transformations. In 1975, Chrzanowski introduced the “outgoing radiation gauge”, a condition on the linearised metric that allows for the construction of the linearised metric from the linearised curvature. In 2019, we proved decay for the metric constructed using Chrzanowski’s outgoing radiation gauge. Recently, using a flow along null geodesics, we have constructed a new gauge such that, in this gauge, the Einstein equation is well posed and such that the linearisation is Chrzanowski’s outgoing radiation gauge.

This is joint work with Lars Andersson, Thomas Backdahl, and Siyuan Ma.

10:30 am–11:30 am

Peter Hintz, MIT (virtual)

Title: Mode stability and shallow quasinormal modes of Kerr-de Sitter black holes

Abstract: The Kerr-de Sitter metric describes a rotating black hole with mass $m$ and specific angular momentum $a$ in a universe, such as our own, with cosmological constant $\Lambda>0$. I will explain a proof of mode stability for the scalar wave equation on Kerr-de Sitter spacetimes in the following setting: fixing $\Lambda$ and the ratio $|a/m|<1$ (related to the subextremality of the black hole in question), mode stability holds for sufficiently small black hole mass $m$. We also obtain estimates for the location of quasinormal modes (resonances) $\sigma$ in any fixed half space $\Im\sigma>-C$. Our results imply that solutions of the wave equation decay exponentially in time to constants, with an explicit exponential rate. The proof is based on careful uniform estimates for the spectral family in the singular limit $m\to 0$ in which, depending on the scaling, the Kerr-de Sitter spacetime limits to a Kerr or the de Sitter spacetime.

11:30 am–12:30 pm

Lars Andersson, Albert Einstein Institute, Germany (virtual)

Title: Gravitational instantons and special geometry

Abstract: Gravitational instantons are Ricci flat complete Riemannian 4-manifolds with at least quadratic curvature decay. In this talk, I will introduce some notions of special geometry, discuss known examples, and mention some open questions. The Chen-Teo gravitational instanton is an asymptotically flat, toric, Ricci flat family of metrics on $\mathrm{CP}^2 \setminus \mathrm{S}^1$, that provides a counterexample to the classical Euclidean Black Hole Uniqueness conjecture. I will sketch a proof that the Chen-Teo Instanton is Hermitian and non-Kähler. Thus, all known examples of gravitational instantons are Hermitian. This talks is based on joint work with Steffen Aksteiner, cf. https://arxiv.org/abs/2112.11863.

12:30 pm–1:30 pm

break

1:30 pm–2:30 pm

Martin Taylor, Imperial College London (virtual)

Title: The nonlinear stability of the Schwarzschild family of black holes

Abstract: I will present a theorem on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes. The proof employs a double null gauge, is expressed entirely in physical space, and utilises the analysis of Dafermos–Holzegel–Rodnianski on the linear stability of the Schwarzschild family. This is joint work with M. Dafermos, G. Holzegel and I. Rodnianski.

2:30 pm–3:30 pm

Po-Ning Chen, University of California, Riverside (virtual)

Title: Angular momentum in general relativity Abstract: The definition of angular momentum in general relativity has been a subtle issue since the 1960s, due to the ‘supertranslation ambiguity’. In this talk, we will discuss how the mathematical theory of quasilocal mass and angular momentum leads to a new definition of angular momentum at null infinity that is free of any supertranslation ambiguity.

This is based on joint work with Jordan Keller, Mu-Tao Wang, Ye-Kai Wang, and Shing-Tung Yau.

3:30 pm–4:00 pm

break

4:00 pm–5:00 pm

Dan Lee, Queens College (CUNY) (hybrid: in person & virtual)

Title: Stability of the positive mass theorem

Abstract: We will discuss the problem of stability for the rigidity part of the Riemannian positive mass theorem, focusing on recent work with Kazaras and Khuri, in which we proved that if one assumes a lower Ricci curvature bound, then stability holds with respect to pointed Gromov-Hausdorff convergence.

Tuesday, April 5, 2022

Time (ET)

Speaker

Title/Abstract

9:30 am–10:30 am

Xinliang An, National University of Singapore (virtual)

Title: Anisotropic dynamical horizons arising in gravitational collapse

Abstract: Black holes are predicted by Einstein’s theory of general relativity, and now we have ample observational evidence for their existence. However theoretically there are many unanswered questions about how black holes come into being and about the structures of their inner spacetime singularities. In this talk, we will present several results in these directions. First, in a joint work with Qing Han, with tools from scale-critical hyperbolic method and non-perturbative elliptic techniques, with anisotropic characteristic initial data we prove that: in the process of gravitational collapse, a smooth and spacelike apparent horizon (dynamical horizon) emerges from general (both isotropic and anisotropic) initial data. This result extends the 2008 Christodoulou’s monumental work and it connects to black hole thermodynamics along the apparent horizon. Second, in joint works with Dejan Gajic and Ruixiang Zhang, for the spherically symmetric Einstein-scalar field system, we derive precise blow-up rates for various geometric quantities along the inner spacelike singularities. These rates obey polynomial blow-up upper bounds; and when it is close to timelike infinity, these rates are not limited to discrete finite choices and they are related to the Price’s law along the event horizon. This indicates a new blow-up phenomenon, driven by a PDE mechanism, rather than an ODE mechanism. If time permits, some results on fluid dynamics will also be addressed.

10:30 am–11:30 am

Sergiu Klainerman, Princeton (virtual)

Title: Nonlinear stability of slowly rotating Kerr solutions

Abstract: I will talk about the status of the stability of Kerr conjecture in General Relativity based on recent results obtained in collaboration with Jeremie Szeftel and Elena Giorgi.

11:30 am–12:30 pm

Siyuan Ma, Sorbonne University (virtual)

Title: Sharp decay for Teukolsky master equation

Abstract: I will talk about joint work with L. Zhang on deriving the late time dynamics of the spin $s$ components that satisfy the Teukolsky master equation in Kerr spacetimes.

12:30 pm–1:30 pm

Break

1:30 pm–2:30 pm

Jonathan Luk, Stanford (virtual)

Title: A tale of two tails

Abstract: Motivated by the strong cosmic censorship conjecture, we introduce a general method for understanding the late-time tail for solutions to wave equations on asymptotically flat spacetimes in odd spatial dimensions. A particular consequence of the method is a re-proof of Price’s law-type results, which concern the sharp decay rate of the late-time tails on stationary spacetimes. Moreover, we show that the late-time tails are in general different from the stationary case in the presence of dynamical and/or nonlinear perturbations. This is a joint work with Sung-Jin Oh (Berkeley).

2:30 pm–3:30 pm

Gary Horowitz, University of California Santa Barbara (virtual)

Title: A new type of extremal black hole

Abstract: I describe a family of four-dimensional, asymptotically flat, charged black holes that develop (charged) scalar hair as one increases their charge at fixed mass. Surprisingly, the maximum charge for given mass is a nonsingular hairy black hole with a nondegenerate event horizon. Since the surface gravity is nonzero, if quantum matter is added, Hawking radiation does not appear to stop when this new extremal limit is reached. This raises the question of whether Hawking radiation will cause the black hole to turn into a naked singularity. I will argue that does not occur.

3:30 pm–4:00 pm

Break

4:00 pm–5:00 pm

Lydia Bieri, University of Michigan (virtual)

Title: Gravitational radiation in general spacetimes

Abstract: Studies of gravitational waves have been devoted mostly to sources such as binary black hole mergers or neutron star mergers, or generally sources that are stationary outside of a compact set. These systems are described by asymptotically-flat manifolds solving the Einstein equations with sufficiently fast decay of the gravitational field towards Minkowski spacetime far away from the source. Waves from such sources have been recorded by the LIGO/VIRGO collaboration since 2015. In this talk, I will present new results on gravitational radiation for sources that are not stationary outside of a compact set, but whose gravitational fields decay more slowly towards infinity. A panorama of new gravitational effects opens up when delving deeper into these more general spacetimes. In particular, whereas the former sources produce memory effects that are finite and of purely electric parity, the latter in addition generate memory of magnetic type, and both types grow. These new effects emerge naturally from the Einstein equations both in the Einstein vacuum case and for neutrino radiation. The latter results are important for sources with extended neutrino halos.

Wednesday, April 6, 2022

Time (ET)

Speaker

Title/Abstract

9:30 am–10:30 am

Gerhard Huisken, Mathematisches Forschungsinstitut Oberwolfach (virtual)

TBA

10:30 am–11:30 am

Carla Cederbaum, Universität Tübingen, Germany (virtual)

Title: Coordinates are messy

Abstract: Asymptotically Euclidean initial data sets $(M,g,K)$ are characterized by the existence of asymptotic coordinates in which the Riemannian metric $g$ and second fundamental form $K$ decay to the Euclidean metric $\delta$ and to $0$ suitably fast, respectively. Provided their matter densities satisfy suitable integrability conditions, they have well-defined (ADM-)energy, (ADM-)linear momentum, and (ADM-)mass. This was proven by Bartnik using harmonic coordinates. To study their (ADM-)angular momentum and (BORT-)center of mass, one usually assumes the existence of Regge—Teitelboim coordinates on the initial data set $(M,g,K)$ in question. We will give examples of asymptotically Euclidean initial data sets which do not possess any Regge—Teitelboim coordinates We will also show that harmonic coordinates can be used as a tool in checking whether a given asymptotically Euclidean initial data set possesses Regge—Teitelboim coordinates. This is joint work with Melanie Graf and Jan Metzger. We will also explain the consequences these findings have for the definition of the center of mass, relying on joint work with Nerz and with Sakovich.

11:30 am–12:30 pm

Stefanos Aretakis, University of Toronto (virtual)

Title: Observational signatures for extremal black holes

Abstract: We will present results regarding the asymptotics of scalar perturbations on black hole backgrounds. We will then derive observational signatures for extremal black holes that are based on global or localized measurements on null infinity. This is based on joint work with Gajic-Angelopoulos and ongoing work with Khanna-Sabharwal.

12:30 pm–1:30 pm

Break

1:30 pm–2:30 pm

Jared Speck, Vanderbilt University (virtual)

Title: The mathematical theory of shock waves in multi-dimensional relativistic and non-relativistic compressible Euler flow

Abstract: In the last two decades, there have been dramatic advances in the rigorous mathematical theory of shock waves in solutions to the relativistic Euler equations and their non-relativistic analog, the compressible Euler equations. A lot of the progress has relied on techniques that were developed to study Einstein’s equations. In this talk, I will provide an overview of the field and highlight some recent progress on problems without symmetry or irrotationality assumptions. I will focus on results that reveal various aspects of the structure of the maximal development of the data and the corresponding implications for the shock development problem, which is the problem of continuing the solution weakly after a shock. I will also describe various open problems, some of which are tied to the Einstein–Euler equations. Various aspects of this program are joint with L. Abbrescia, M. Disconzi, and J. Luk.

2:30 pm–3:30 pm

Lan-Hsuan Huang, University of Connecticut (hybrid: in person & virtual)

Title: Null perfect fluids, improvability of dominant energy scalar, and Bartnik mass minimizers

Abstract: We introduce the concept of improvability of the dominant energy scalar, and we derive strong consequences of non-improvability. In particular, we prove that a non-improvable initial data set without local symmetries must sit inside a null perfect fluid spacetime carrying a global Killing vector field. We also show that the dominant energy scalar is always almost improvable in a precise sense. Using these main results, we provide a characterization of Bartnik mass minimizing initial data sets which makes substantial progress toward Bartnik’s stationary conjecture.

Along the way we observe that in dimensions greater than eight there exist pp-wave counterexamples (without the optimal decay rate for asymptotically flatness) to the equality case of the spacetime positive mass theorem. As a consequence, we find counterexamples to Bartnik’s stationary and strict positivity conjectures in those dimensions. This talk is based on joint work with Dan A. Lee.

3:30 pm–4:00 pm

Break

4:00 pm–5:00 pm

Demetre Kazaras, Duke University (virtual)

Title: Comparison geometry for scalar curvature and spacetime harmonic functions

Abstract: Comparison theorems are the basis for our geometric understanding of Riemannian manifolds satisfying a given curvature condition. A remarkable example is the Gromov-Lawson toric band inequality, which bounds the distance between the two sides of a Riemannian torus-cross-interval with positive scalar curvature by a sharp constant inversely proportional to the scalar curvature’s minimum. We will give a new qualitative version of this and similar band-type inequalities in dimension 3 using the notion of spacetime harmonic functions, which recently played the lead role in our recent proof of the positive mass theorem. This is joint work with Sven Hirsch, Marcus Khuri, and Yiyue Zhang.

Thursday, April 7, 2022

Time (ET)

Speaker

Title/Abstract

9:30 am–10:30 am

Piotr Chrusciel, Universitat Wien (virtual)

Title: Maskit gluing and hyperbolic mass

Abstract: “Maskit gluing” is a gluing construction for asymptotically locally hyperbolic (ALH) manifolds with negative cosmological constant. I will present a formula for the mass of Maskit-glued ALH manifolds and describe how it can be used to construct general relativistic initial data with negative mass.

10:30 am–11:30 am

Greg Galloway, University of Miami (virtual)

Title: Initial data rigidity and applications

Abstract: We present a result from our work with Michael Eichmair and Abraão Mendes concerning initial data rigidity results (CMP, 2021), and look at some consequences. In a note with Piotr Chruściel (CQG 2021), we showed how to use this result, together with arguments from Chruściel and Delay’s proof of the their hyperbolic PMT result, to obtain a hyperbolic PMT result with boundary. This will also be discussed.

11:30 am–12:30 pm

Pengzi Miao, University of Miami (virtual)

Title: Some remarks on mass and quasi-local mass

Abstract: In the first part of this talk, I will describe how to detect the mass of asymptotically flat and asymptotically hyperbolic manifolds via large Riemannian polyhedra. In the second part, I will discuss an estimate of the Bartnik quasi-local mass and its geometric implications. This talk is based on several joint works with A. Piubello, and with H.C. Jang.

12:30 pm–1:30 pm

Break

1:30 pm–2:30 pm

Yakov Shlapentokh Rothman, Princeton (hybrid: in person & virtual)

Title: Self-Similarity and Naked Singularities for the Einstein Vacuum Equations

Abstract: We will start with an introduction to the problem of constructing naked singularities for the Einstein vacuum equations, and then explain our discovery of a fundamentally new type of self-similarity and show how this allows us to construct solutions corresponding to a naked singularity. This is joint work with Igor Rodnianski.

2:30 pm–3:30 pm

Marcelo Disconzi, Vanderbilt University (virtual)

Title: General-relativistic viscous fluids.

Abstract: The discovery of the quark-gluon plasma that forms in heavy-ion collision experiments provides a unique opportunity to study the properties of matter under extreme conditions, as the quark-gluon plasma is the hottest, smallest, and densest fluid known to humanity. Studying the quark-gluon plasma also provides a window into the earliest moments of the universe, since microseconds after the Big Bang the universe was filled with matter in the form of the quark-gluon plasma. For more than two decades, the community has intensely studied the quark-gluon plasma with the help of a rich interaction between experiments, theory, phenomenology, and numerical simulations. From these investigations, a coherent picture has emerged, indicating that the quark-gluon plasma behaves essentially like a relativistic liquid with viscosity. More recently, state-of-the-art numerical simulations strongly suggested that viscous and dissipative effects can also have non-negligible effects on gravitational waves produced by binary neutron star mergers. But despite the importance of viscous effects for the study of such systems, a robust and mathematically sound theory of relativistic fluids with viscosity is still lacking. This is due, in part, to difficulties to preserve causality upon the inclusion of viscous and dissipative effects into theories of relativistic fluids. In this talk, we will survey the history of the problem and report on a new approach to relativistic viscous fluids that addresses these issues.

3:30 pm–4:00 pm

Break

4:00 pm–5:00 pm

Maxime van de Moortel, Princeton (hybrid: in person & virtual)

Title: Black holes: the inside story of gravitational collapse

Abstract: What is inside a dynamical black hole? While the local region near time-like infinity is understood for various models, the global structure of the black hole interior has largely remained unexplored. These questions are deeply connected to the nature of singularities in General Relativity and celebrated problems such as Penrose’s Strong Cosmic Censorship Conjecture. I will present my recent resolution of these problems in spherical gravitational collapse, based on the discovery of a novel phenomenon: the breakdown of weak singularities and the dynamical formation of a strong singularity.

Friday, April 8, 2022

Time (ET)

Speaker

Title/Abstract

9:30 am–10:30 am

Ye-Kai Wang, National Cheng Kun University, Taiwan (virtual)

Title: Supertranslation invariance of angular momentum at null infinity in double null gauge

Abstract: This talk accompanies Po-Ning Chen’s talk on Monday with the results described in the double null gauge rather than Bondi-Sachs coordinates. Besides discussing how Chen-Wang-Yau angular momentum resolves the supertranslation ambiguity, we also review the definition of angular momentum defined by A. Rizzi. The talk is based on the joint work with Po-Ning Chen, Jordan Keller, Mu-Tao Wang, and Shing-Tung Yau.

10:30 am–11:30 am

Zoe Wyatt, King’s College London (virtual)

Title: Global Stability of Spacetimes with Supersymmetric Compactifications

Abstract: Spacetimes with compact directions which have special holonomy, such as Calabi-Yau spaces, play an important role in supergravity and string theory. In this talk I will discuss a recent work with Lars Andersson, Pieter Blue and Shing-Tung Yau, where we show the global, nonlinear stability a spacetime which is a cartesian product of a high dimensional Minkowski space with a compact Ricci flat internal space with special holonomy. This stability result is related to a conjecture of Penrose concerning the validity of string theory. Our proof uses the intersection of methods for quasilinear wave and Klein-Gordon equations, and so towards the end of the talk I will also comment more generally on coupled wave–Klein-Gordon equations.

11:30 am–12:30 pm

Elena Giorgi, Columbia University (hybrid: in person & virtual)

Title: The stability of charged black holes

Abstract: Black hole solutions in General Relativity are parametrized by their mass, spin and charge. In this talk, I will motivate why the charge of black holes adds interesting dynamics to solutions of the Einstein equation thanks to the interaction between gravitational and electromagnetic radiation. Such radiations are solutions of a system of coupled wave equations with a symmetric structure which allows to define a combined energy-momentum tensor for the system. Finally, I will show how this physical-space approach is resolutive in the most general case of Kerr-Newman black hole, where the interaction between the radiations prevents the separability in modes.

12:30 pm–1:30 pm

Break

1:30 pm–2:30 pm

Marcus Khuri, Stony Brook University (virtual)

Title: The mass-angular momentum inequality for multiple black holes Abstract: Consider a complete 3-dimensional initial data set for the Einstein equations which has multiple asymptotically flat or asymptotically cylindrical ends. If it is simply connected, axisymmetric, maximal, and satisfies the appropriate energy condition then the ADM mass of any of the asymptotically flat ends is bounded below by the square root of the total angular momentum. This generalizes previous work of Dain, Chrusciel-Li-Weinstein, and Schoen-Zhou which treated either the single black hole case or the multiple black hole case without an explicit lower bound. The proof relies on an analysis of the asymptotics of singular harmonic maps from R^3 \ \Gamma –>H^2 where \Gamma is a coordinate axis. This is joint work with Q. Han, G. Weinstein, and J. Xiong.

2:30 pm–3:30 pm

Martin Lesourd, Harvard (hybrid: in person & virtual)

Title: A Snippet on Mass and the Topology and Geometry of Positive Scalar Curvature

Abstract: I will talk about a small corner of the study of Positive Scalar Curvature (PSC) and questions which are most closely related to the Positive Mass Theorem. The classic questions are ”which topologies allow for PSC?” and ”what is the geometry of manifolds with PSC?”. This is based on joint work with Prof. S-T. Yau, Prof. D. A. Lee, and R. Unger.

3:30 pm–4:00 pm

Break

4:00 pm–5:00 pm

Georgios Moschidis, Princeton (virtual)

Title: Weak turbulence for the Einstein–scalar field system.

Abstract: In the presence of confinement, the Einstein field equations are expected to exhibit turbulent dynamics. In the presence of a negative cosmological constant, the AdS instability conjecture claims the existence of arbitrarily small perturbations to the initial data of Anti-de Sitter spacetime which, under evolution by the vacuum Einstein equations with reflecting boundary conditions at conformal infinity, lead to the formation of black holes after sufficiently long time. In this talk, I will present a rigorous proof of the AdS instability conjecture in the setting of the spherically symmetric Einstein-scalar field system. The construction of the unstable initial data will require carefully designing a family of initial configurations of localized matter beams and estimating the exchange of energy taking place between interacting beams over long periods of time, as well as estimating the decoherence rate of those beams.

For the full schedule, please see: https://cmsa.fas.harvard.edu/gr-program/

This conference will be held virtually on Zoom. Registration is required. Webinar Registration

A few talks will be held in hybrid formats, with talks given from the CMSA seminar room, G-10. Advanced registration for in-person components is required. In-Person Registration

For more information, please see https://cmsa.fas.harvard.edu/gr-program/

The Harvard CMSA will be hosting a conference on General Relativity from April 4-8, 2022.

This conference will be held virtually on Zoom. Registration is required. Webinar Registration A few talks will be held in hybrid formats, with talks given from the CMSA seminar room, G-10. Advanced registration for in-person components is required. In-Person Registration

For more information, please see: https://cmsa.fas.harvard.edu/gr-program/

Are we alone? It would be arrogant to think that we are, given that a quarter of all stars host a habitable Earth-size planet. Upcoming searches will aim to detect markers of life in the atmospheres of planets outside the Solar System. We also have unprecedented technologies to detect signs of intelligent civilizations through industrial pollution of planetary atmospheres, space archaeology of debris from dead civilizations or artifacts such as photovoltaic cells that are used to re-distribute light and heat on the surface of a planet or giant megastructures. Our own civilization is starting to explore interstellar travel. Essential information may also arrive as a “message in a bottle”, implying that we should examine carefully any unusual object that arrives to our vicinity from outside the Solar System, such as `Oumuamua.

Bio: Abraham (Avi) Loeb is the Frank B. Baird, Jr., Professor of Science at Harvard University and a bestselling author (in lists of the New York Times, Wall Street Journal, Publishers Weekly, Die Zeit, Der Spiegel, L’Express and more). He received a PhD in Physics from the Hebrew University of Jerusalem in Israel at age 24 (1980–1986), led the first international project supported by the Strategic Defense Initiative (1983–1988), and was subsequently a long-term member of the Institute for Advanced Study at Princeton (1988–1993). Loeb has written 8 books, including most recently, Extraterrestrial (Houghton Mifflin Harcourt, 2021), and nearly a thousand papers (with an h-index of 118) on a wide range of topics, including black holes, the first stars, the search for extraterrestrial life, and the future of the Universe. Loeb is the head of the Galileo Project in search for extraterrestrial intelligence, the Director of the Institute for Theory and Computation (2007–present) within the Harvard-Smithsonian Center for Astrophysics, and also serves as the Head of the Galileo Project (2021–present). He had been the longest serving Chair of Harvard’s Department of Astronomy (2011–2020) and the Founding Director of Harvard’s Black Hole Initiative (2016–2021). He is an elected fellow of the American Academy of Arts & Sciences, the American Physical Society, and the International Academy of Astronautics. Loeb is a former member of the President’s Council of Advisors on Science and Technology (PCAST) at the White House, a former chair of the Board on Physics and Astronomy of the National Academies (2018–2021) and a current member of the Advisory Board for “Einstein: Visualize the Impossible” of the Hebrew University. He also chairs the Advisory Committee for the Breakthrough Starshot Initiative (2016–present) and serves as the Science Theory Director for all Initiatives of the Breakthrough Prize Foundation. In 2012, TIME magazine selected Loeb as one of the 25 most influential people in space and in 2020 Loeb was selected among the 14 most inspiring Israelis of the last decade.

Click here for Loeb’s commentaries on innovation and diversity.

Refreshments will be served between 6:30–7:00 pm before the lecture.

Pieter Blue, University of Edinburgh, UK (virtual)

Title: Linear stability of the Kerr spacetime in the outgoing radiation gauge

Abstract: This talk will discuss a new gauge condition (i.e. coordinate condition) for the Einstein equation, the linearisation of the Einstein equation in this gauge, and the decay of solutions to the linearised Einstein equation around Kerr black holes in this gauge. The stability of the family of Kerr black holes under the evolution generated by the Einstein equation is a long-standing problem in mathematical relativity. In 1972, Teukolsky discovered equations governing certain components of the linearised curvature that are invariant under linearised gague transformations. In 1975, Chrzanowski introduced the “outgoing radiation gauge”, a condition on the linearised metric that allows for the construction of the linearised metric from the linearised curvature. In 2019, we proved decay for the metric constructed using Chrzanowski’s outgoing radiation gauge. Recently, using a flow along null geodesics, we have constructed a new gauge such that, in this gauge, the Einstein equation is well posed and such that the linearisation is Chrzanowski’s outgoing radiation gauge.

This is joint work with Lars Andersson, Thomas Backdahl, and Siyuan Ma.

10:30 am–11:30 am

Peter Hintz, MIT (virtual)

Title: Mode stability and shallow quasinormal modes of Kerr-de Sitter black holes

Abstract: The Kerr-de Sitter metric describes a rotating black hole with mass $m$ and specific angular momentum $a$ in a universe, such as our own, with cosmological constant $\Lambda>0$. I will explain a proof of mode stability for the scalar wave equation on Kerr-de Sitter spacetimes in the following setting: fixing $\Lambda$ and the ratio $|a/m|<1$ (related to the subextremality of the black hole in question), mode stability holds for sufficiently small black hole mass $m$. We also obtain estimates for the location of quasinormal modes (resonances) $\sigma$ in any fixed half space $\Im\sigma>-C$. Our results imply that solutions of the wave equation decay exponentially in time to constants, with an explicit exponential rate. The proof is based on careful uniform estimates for the spectral family in the singular limit $m\to 0$ in which, depending on the scaling, the Kerr-de Sitter spacetime limits to a Kerr or the de Sitter spacetime.

11:30 am–12:30 pm

Lars Andersson, Albert Einstein Institute, Germany (virtual)

Title: Gravitational instantons and special geometry

Abstract: Gravitational instantons are Ricci flat complete Riemannian 4-manifolds with at least quadratic curvature decay. In this talk, I will introduce some notions of special geometry, discuss known examples, and mention some open questions. The Chen-Teo gravitational instanton is an asymptotically flat, toric, Ricci flat family of metrics on $\mathrm{CP}^2 \setminus \mathrm{S}^1$, that provides a counterexample to the classical Euclidean Black Hole Uniqueness conjecture. I will sketch a proof that the Chen-Teo Instanton is Hermitian and non-Kähler. Thus, all known examples of gravitational instantons are Hermitian. This talks is based on joint work with Steffen Aksteiner, cf. https://arxiv.org/abs/2112.11863.

12:30 pm–1:30 pm

break

1:30 pm–2:30 pm

Martin Taylor, Imperial College London (virtual)

Title: The nonlinear stability of the Schwarzschild family of black holes

Abstract: I will present a theorem on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes. The proof employs a double null gauge, is expressed entirely in physical space, and utilises the analysis of Dafermos–Holzegel–Rodnianski on the linear stability of the Schwarzschild family. This is joint work with M. Dafermos, G. Holzegel and I. Rodnianski.

2:30 pm–3:30 pm

Po-Ning Chen, University of California, Riverside (virtual)

Title: Angular momentum in general relativity Abstract: The definition of angular momentum in general relativity has been a subtle issue since the 1960s, due to the ‘supertranslation ambiguity’. In this talk, we will discuss how the mathematical theory of quasilocal mass and angular momentum leads to a new definition of angular momentum at null infinity that is free of any supertranslation ambiguity.

This is based on joint work with Jordan Keller, Mu-Tao Wang, Ye-Kai Wang, and Shing-Tung Yau.

3:30 pm–4:00 pm

break

4:00 pm–5:00 pm

Dan Lee, Queens College (CUNY) (hybrid: in person & virtual)

Title: Stability of the positive mass theorem

Abstract: We will discuss the problem of stability for the rigidity part of the Riemannian positive mass theorem, focusing on recent work with Kazaras and Khuri, in which we proved that if one assumes a lower Ricci curvature bound, then stability holds with respect to pointed Gromov-Hausdorff convergence.

Tuesday, April 5, 2022

Time (ET)

Speaker

Title/Abstract

9:30 am–10:30 am

Xinliang An, National University of Singapore (virtual)

Title: Anisotropic dynamical horizons arising in gravitational collapse

Abstract: Black holes are predicted by Einstein’s theory of general relativity, and now we have ample observational evidence for their existence. However theoretically there are many unanswered questions about how black holes come into being and about the structures of their inner spacetime singularities. In this talk, we will present several results in these directions. First, in a joint work with Qing Han, with tools from scale-critical hyperbolic method and non-perturbative elliptic techniques, with anisotropic characteristic initial data we prove that: in the process of gravitational collapse, a smooth and spacelike apparent horizon (dynamical horizon) emerges from general (both isotropic and anisotropic) initial data. This result extends the 2008 Christodoulou’s monumental work and it connects to black hole thermodynamics along the apparent horizon. Second, in joint works with Dejan Gajic and Ruixiang Zhang, for the spherically symmetric Einstein-scalar field system, we derive precise blow-up rates for various geometric quantities along the inner spacelike singularities. These rates obey polynomial blow-up upper bounds; and when it is close to timelike infinity, these rates are not limited to discrete finite choices and they are related to the Price’s law along the event horizon. This indicates a new blow-up phenomenon, driven by a PDE mechanism, rather than an ODE mechanism. If time permits, some results on fluid dynamics will also be addressed.

10:30 am–11:30 am

Sergiu Klainerman, Princeton (virtual)

Title: Nonlinear stability of slowly rotating Kerr solutions

Abstract: I will talk about the status of the stability of Kerr conjecture in General Relativity based on recent results obtained in collaboration with Jeremie Szeftel and Elena Giorgi.

11:30 am–12:30 pm

Siyuan Ma, Sorbonne University (virtual)

Title: Sharp decay for Teukolsky master equation

Abstract: I will talk about joint work with L. Zhang on deriving the late time dynamics of the spin $s$ components that satisfy the Teukolsky master equation in Kerr spacetimes.

12:30 pm–1:30 pm

Break

1:30 pm–2:30 pm

Jonathan Luk, Stanford (virtual)

Title: A tale of two tails

Abstract: Motivated by the strong cosmic censorship conjecture, we introduce a general method for understanding the late-time tail for solutions to wave equations on asymptotically flat spacetimes in odd spatial dimensions. A particular consequence of the method is a re-proof of Price’s law-type results, which concern the sharp decay rate of the late-time tails on stationary spacetimes. Moreover, we show that the late-time tails are in general different from the stationary case in the presence of dynamical and/or nonlinear perturbations. This is a joint work with Sung-Jin Oh (Berkeley).

2:30 pm–3:30 pm

Gary Horowitz, University of California Santa Barbara (virtual)

Title: A new type of extremal black hole

Abstract: I describe a family of four-dimensional, asymptotically flat, charged black holes that develop (charged) scalar hair as one increases their charge at fixed mass. Surprisingly, the maximum charge for given mass is a nonsingular hairy black hole with a nondegenerate event horizon. Since the surface gravity is nonzero, if quantum matter is added, Hawking radiation does not appear to stop when this new extremal limit is reached. This raises the question of whether Hawking radiation will cause the black hole to turn into a naked singularity. I will argue that does not occur.

3:30 pm–4:00 pm

Break

4:00 pm–5:00 pm

Lydia Bieri, University of Michigan (virtual)

Title: Gravitational radiation in general spacetimes

Abstract: Studies of gravitational waves have been devoted mostly to sources such as binary black hole mergers or neutron star mergers, or generally sources that are stationary outside of a compact set. These systems are described by asymptotically-flat manifolds solving the Einstein equations with sufficiently fast decay of the gravitational field towards Minkowski spacetime far away from the source. Waves from such sources have been recorded by the LIGO/VIRGO collaboration since 2015. In this talk, I will present new results on gravitational radiation for sources that are not stationary outside of a compact set, but whose gravitational fields decay more slowly towards infinity. A panorama of new gravitational effects opens up when delving deeper into these more general spacetimes. In particular, whereas the former sources produce memory effects that are finite and of purely electric parity, the latter in addition generate memory of magnetic type, and both types grow. These new effects emerge naturally from the Einstein equations both in the Einstein vacuum case and for neutrino radiation. The latter results are important for sources with extended neutrino halos.

Wednesday, April 6, 2022

Time (ET)

Speaker

Title/Abstract

9:30 am–10:30 am

Gerhard Huisken, Mathematisches Forschungsinstitut Oberwolfach (virtual)

TBA

10:30 am–11:30 am

Carla Cederbaum, Universität Tübingen, Germany (virtual)

Title: Coordinates are messy

Abstract: Asymptotically Euclidean initial data sets $(M,g,K)$ are characterized by the existence of asymptotic coordinates in which the Riemannian metric $g$ and second fundamental form $K$ decay to the Euclidean metric $\delta$ and to $0$ suitably fast, respectively. Provided their matter densities satisfy suitable integrability conditions, they have well-defined (ADM-)energy, (ADM-)linear momentum, and (ADM-)mass. This was proven by Bartnik using harmonic coordinates. To study their (ADM-)angular momentum and (BORT-)center of mass, one usually assumes the existence of Regge—Teitelboim coordinates on the initial data set $(M,g,K)$ in question. We will give examples of asymptotically Euclidean initial data sets which do not possess any Regge—Teitelboim coordinates We will also show that harmonic coordinates can be used as a tool in checking whether a given asymptotically Euclidean initial data set possesses Regge—Teitelboim coordinates. This is joint work with Melanie Graf and Jan Metzger. We will also explain the consequences these findings have for the definition of the center of mass, relying on joint work with Nerz and with Sakovich.

11:30 am–12:30 pm

Stefanos Aretakis, University of Toronto (virtual)

Title: Observational signatures for extremal black holes

Abstract: We will present results regarding the asymptotics of scalar perturbations on black hole backgrounds. We will then derive observational signatures for extremal black holes that are based on global or localized measurements on null infinity. This is based on joint work with Gajic-Angelopoulos and ongoing work with Khanna-Sabharwal.

12:30 pm–1:30 pm

Break

1:30 pm–2:30 pm

Jared Speck, Vanderbilt University (virtual)

Title: The mathematical theory of shock waves in multi-dimensional relativistic and non-relativistic compressible Euler flow

Abstract: In the last two decades, there have been dramatic advances in the rigorous mathematical theory of shock waves in solutions to the relativistic Euler equations and their non-relativistic analog, the compressible Euler equations. A lot of the progress has relied on techniques that were developed to study Einstein’s equations. In this talk, I will provide an overview of the field and highlight some recent progress on problems without symmetry or irrotationality assumptions. I will focus on results that reveal various aspects of the structure of the maximal development of the data and the corresponding implications for the shock development problem, which is the problem of continuing the solution weakly after a shock. I will also describe various open problems, some of which are tied to the Einstein–Euler equations. Various aspects of this program are joint with L. Abbrescia, M. Disconzi, and J. Luk.

2:30 pm–3:30 pm

Lan-Hsuan Huang, University of Connecticut (hybrid: in person & virtual)

Title: Null perfect fluids, improvability of dominant energy scalar, and Bartnik mass minimizers

Abstract: We introduce the concept of improvability of the dominant energy scalar, and we derive strong consequences of non-improvability. In particular, we prove that a non-improvable initial data set without local symmetries must sit inside a null perfect fluid spacetime carrying a global Killing vector field. We also show that the dominant energy scalar is always almost improvable in a precise sense. Using these main results, we provide a characterization of Bartnik mass minimizing initial data sets which makes substantial progress toward Bartnik’s stationary conjecture.

Along the way we observe that in dimensions greater than eight there exist pp-wave counterexamples (without the optimal decay rate for asymptotically flatness) to the equality case of the spacetime positive mass theorem. As a consequence, we find counterexamples to Bartnik’s stationary and strict positivity conjectures in those dimensions. This talk is based on joint work with Dan A. Lee.

3:30 pm–4:00 pm

Break

4:00 pm–5:00 pm

Demetre Kazaras, Duke University (virtual)

Title: Comparison geometry for scalar curvature and spacetime harmonic functions

Abstract: Comparison theorems are the basis for our geometric understanding of Riemannian manifolds satisfying a given curvature condition. A remarkable example is the Gromov-Lawson toric band inequality, which bounds the distance between the two sides of a Riemannian torus-cross-interval with positive scalar curvature by a sharp constant inversely proportional to the scalar curvature’s minimum. We will give a new qualitative version of this and similar band-type inequalities in dimension 3 using the notion of spacetime harmonic functions, which recently played the lead role in our recent proof of the positive mass theorem. This is joint work with Sven Hirsch, Marcus Khuri, and Yiyue Zhang.

Thursday, April 7, 2022

Time (ET)

Speaker

Title/Abstract

9:30 am–10:30 am

Piotr Chrusciel, Universitat Wien (virtual)

Title: Maskit gluing and hyperbolic mass

Abstract: “Maskit gluing” is a gluing construction for asymptotically locally hyperbolic (ALH) manifolds with negative cosmological constant. I will present a formula for the mass of Maskit-glued ALH manifolds and describe how it can be used to construct general relativistic initial data with negative mass.

10:30 am–11:30 am

Greg Galloway, University of Miami (virtual)

Title: Initial data rigidity and applications

Abstract: We present a result from our work with Michael Eichmair and Abraão Mendes concerning initial data rigidity results (CMP, 2021), and look at some consequences. In a note with Piotr Chruściel (CQG 2021), we showed how to use this result, together with arguments from Chruściel and Delay’s proof of the their hyperbolic PMT result, to obtain a hyperbolic PMT result with boundary. This will also be discussed.

11:30 am–12:30 pm

Pengzi Miao, University of Miami (virtual)

Title: Some remarks on mass and quasi-local mass

Abstract: In the first part of this talk, I will describe how to detect the mass of asymptotically flat and asymptotically hyperbolic manifolds via large Riemannian polyhedra. In the second part, I will discuss an estimate of the Bartnik quasi-local mass and its geometric implications. This talk is based on several joint works with A. Piubello, and with H.C. Jang.

12:30 pm–1:30 pm

Break

1:30 pm–2:30 pm

Yakov Shlapentokh Rothman, Princeton (hybrid: in person & virtual)

Title: Self-Similarity and Naked Singularities for the Einstein Vacuum Equations

Abstract: We will start with an introduction to the problem of constructing naked singularities for the Einstein vacuum equations, and then explain our discovery of a fundamentally new type of self-similarity and show how this allows us to construct solutions corresponding to a naked singularity. This is joint work with Igor Rodnianski.

2:30 pm–3:30 pm

Marcelo Disconzi, Vanderbilt University (virtual)

Title: General-relativistic viscous fluids.

Abstract: The discovery of the quark-gluon plasma that forms in heavy-ion collision experiments provides a unique opportunity to study the properties of matter under extreme conditions, as the quark-gluon plasma is the hottest, smallest, and densest fluid known to humanity. Studying the quark-gluon plasma also provides a window into the earliest moments of the universe, since microseconds after the Big Bang the universe was filled with matter in the form of the quark-gluon plasma. For more than two decades, the community has intensely studied the quark-gluon plasma with the help of a rich interaction between experiments, theory, phenomenology, and numerical simulations. From these investigations, a coherent picture has emerged, indicating that the quark-gluon plasma behaves essentially like a relativistic liquid with viscosity. More recently, state-of-the-art numerical simulations strongly suggested that viscous and dissipative effects can also have non-negligible effects on gravitational waves produced by binary neutron star mergers. But despite the importance of viscous effects for the study of such systems, a robust and mathematically sound theory of relativistic fluids with viscosity is still lacking. This is due, in part, to difficulties to preserve causality upon the inclusion of viscous and dissipative effects into theories of relativistic fluids. In this talk, we will survey the history of the problem and report on a new approach to relativistic viscous fluids that addresses these issues.

3:30 pm–4:00 pm

Break

4:00 pm–5:00 pm

Maxime van de Moortel, Princeton (hybrid: in person & virtual)

Title: Black holes: the inside story of gravitational collapse

Abstract: What is inside a dynamical black hole? While the local region near time-like infinity is understood for various models, the global structure of the black hole interior has largely remained unexplored. These questions are deeply connected to the nature of singularities in General Relativity and celebrated problems such as Penrose’s Strong Cosmic Censorship Conjecture. I will present my recent resolution of these problems in spherical gravitational collapse, based on the discovery of a novel phenomenon: the breakdown of weak singularities and the dynamical formation of a strong singularity.

Friday, April 8, 2022

Time (ET)

Speaker

Title/Abstract

9:30 am–10:30 am

Ye-Kai Wang, National Cheng Kun University, Taiwan (virtual)

Title: Supertranslation invariance of angular momentum at null infinity in double null gauge

Abstract: This talk accompanies Po-Ning Chen’s talk on Monday with the results described in the double null gauge rather than Bondi-Sachs coordinates. Besides discussing how Chen-Wang-Yau angular momentum resolves the supertranslation ambiguity, we also review the definition of angular momentum defined by A. Rizzi. The talk is based on the joint work with Po-Ning Chen, Jordan Keller, Mu-Tao Wang, and Shing-Tung Yau.

10:30 am–11:30 am

Zoe Wyatt, King’s College London (virtual)

Title: Global Stability of Spacetimes with Supersymmetric Compactifications

Abstract: Spacetimes with compact directions which have special holonomy, such as Calabi-Yau spaces, play an important role in supergravity and string theory. In this talk I will discuss a recent work with Lars Andersson, Pieter Blue and Shing-Tung Yau, where we show the global, nonlinear stability a spacetime which is a cartesian product of a high dimensional Minkowski space with a compact Ricci flat internal space with special holonomy. This stability result is related to a conjecture of Penrose concerning the validity of string theory. Our proof uses the intersection of methods for quasilinear wave and Klein-Gordon equations, and so towards the end of the talk I will also comment more generally on coupled wave–Klein-Gordon equations.

11:30 am–12:30 pm

Elena Giorgi, Columbia University (hybrid: in person & virtual)

Title: The stability of charged black holes

Abstract: Black hole solutions in General Relativity are parametrized by their mass, spin and charge. In this talk, I will motivate why the charge of black holes adds interesting dynamics to solutions of the Einstein equation thanks to the interaction between gravitational and electromagnetic radiation. Such radiations are solutions of a system of coupled wave equations with a symmetric structure which allows to define a combined energy-momentum tensor for the system. Finally, I will show how this physical-space approach is resolutive in the most general case of Kerr-Newman black hole, where the interaction between the radiations prevents the separability in modes.

12:30 pm–1:30 pm

Break

1:30 pm–2:30 pm

Marcus Khuri, Stony Brook University (virtual)

Title: The mass-angular momentum inequality for multiple black holes Abstract: Consider a complete 3-dimensional initial data set for the Einstein equations which has multiple asymptotically flat or asymptotically cylindrical ends. If it is simply connected, axisymmetric, maximal, and satisfies the appropriate energy condition then the ADM mass of any of the asymptotically flat ends is bounded below by the square root of the total angular momentum. This generalizes previous work of Dain, Chrusciel-Li-Weinstein, and Schoen-Zhou which treated either the single black hole case or the multiple black hole case without an explicit lower bound. The proof relies on an analysis of the asymptotics of singular harmonic maps from R^3 \ \Gamma –>H^2 where \Gamma is a coordinate axis. This is joint work with Q. Han, G. Weinstein, and J. Xiong.

2:30 pm–3:30 pm

Martin Lesourd, Harvard (hybrid: in person & virtual)

Title: A Snippet on Mass and the Topology and Geometry of Positive Scalar Curvature

Abstract: I will talk about a small corner of the study of Positive Scalar Curvature (PSC) and questions which are most closely related to the Positive Mass Theorem. The classic questions are ”which topologies allow for PSC?” and ”what is the geometry of manifolds with PSC?”. This is based on joint work with Prof. S-T. Yau, Prof. D. A. Lee, and R. Unger.

3:30 pm–4:00 pm

Break

4:00 pm–5:00 pm

Georgios Moschidis, Princeton (virtual)

Title: Weak turbulence for the Einstein–scalar field system.

Abstract: In the presence of confinement, the Einstein field equations are expected to exhibit turbulent dynamics. In the presence of a negative cosmological constant, the AdS instability conjecture claims the existence of arbitrarily small perturbations to the initial data of Anti-de Sitter spacetime which, under evolution by the vacuum Einstein equations with reflecting boundary conditions at conformal infinity, lead to the formation of black holes after sufficiently long time. In this talk, I will present a rigorous proof of the AdS instability conjecture in the setting of the spherically symmetric Einstein-scalar field system. The construction of the unstable initial data will require carefully designing a family of initial configurations of localized matter beams and estimating the exchange of energy taking place between interacting beams over long periods of time, as well as estimating the decoherence rate of those beams.

For the full schedule, please see: https://cmsa.fas.harvard.edu/gr-program/

This conference will be held virtually on Zoom. Registration is required. Webinar Registration

A few talks will be held in hybrid formats, with talks given from the CMSA seminar room, G-10. Advanced registration for in-person components is required. In-Person Registration

For more information, please see https://cmsa.fas.harvard.edu/gr-program/

I will start the lecture with a brief introduction to various flavors of the Langlands correspondence. I will then explain the setup of a new flavor: the analytic Langlands correspondence for complex curves which was recently fomulated by Pavel Etingof, David Kazhdan, and myself (see arXiv:1908.09677, 2103.01509, and 2106.0524). It can be interpreted in terms of a quantum integrable system, which is obtained by “doubling” the celebrated quantum Hitchin system.

I will report a recent approach of regularizing divergent integrals on configuration spaces of Riemann surfaces, introduced by Si Li and myself in arXiv:2008.07503, with an emphasis on genus one cases where modular forms arise naturally. I will then talk about some applications in studying correlation functions in 2d chiral CFTs, holomorphic anomaly equations, etc. If time permits, I will also mention a more algebraic formulation of this notion of regularized integrals in terms of mixed Hodge structures.

For information on how to join, please see: https://cmsa.fas.harvard.edu/seminars-and-colloquium/

The Harvard CMSA will be hosting a conference on General Relativity from April 4-8, 2022.

This conference will be held virtually on Zoom. Registration is required. Webinar Registration A few talks will be held in hybrid formats, with talks given from the CMSA seminar room, G-10. Advanced registration for in-person components is required. In-Person Registration

For more information, please see: https://cmsa.fas.harvard.edu/gr-program/

The universal centralizer of a complex semisimple adjoint group G is the family of regular centralizers in G, parametrized by the regular conjugacy classes. It has a natural symplectic structure which is inherited from the cotangent bundle of G. I will construct a smooth, log-symplectic relative compactification of this family using the wonderful compactification of G. Its compactified centralizer fibers are isomorphic to Hessenberg varieties, and its symplectic leaves are indexed by root system combinatorics. I will also explain how to produce a multiplicative analogue of this construction, by moving from the Poisson to the quasi-Poisson setting.

Gromov-Witten invariants of holomorphic symplectic 4-folds vanish and one can consider the corresponding reduced theory. In this talk, we will explain a definition of Gopakumar-Vafa type invariants for such a reduced theory. These invariants are conjectured to be integers and have alternative interpretations using sheaf theoretic moduli spaces. Our conjecture is proved for the product of two K3 surfaces, which naturally leads to a closed formula of Fujiki constants of Chern classes of tangent bundles of Hilbert schemes of points on K3 surfaces. On a very general holomorphic symplectic 4-folds of K3^[2] type, our conjecture provides a Yau-Zaslow type formula for the number of isolated genus 2 curves of minimal degree. Based on joint works with Georg Oberdieck and Yukinobu Toda.

Pieter Blue, University of Edinburgh, UK (virtual)

Title: Linear stability of the Kerr spacetime in the outgoing radiation gauge

Abstract: This talk will discuss a new gauge condition (i.e. coordinate condition) for the Einstein equation, the linearisation of the Einstein equation in this gauge, and the decay of solutions to the linearised Einstein equation around Kerr black holes in this gauge. The stability of the family of Kerr black holes under the evolution generated by the Einstein equation is a long-standing problem in mathematical relativity. In 1972, Teukolsky discovered equations governing certain components of the linearised curvature that are invariant under linearised gague transformations. In 1975, Chrzanowski introduced the “outgoing radiation gauge”, a condition on the linearised metric that allows for the construction of the linearised metric from the linearised curvature. In 2019, we proved decay for the metric constructed using Chrzanowski’s outgoing radiation gauge. Recently, using a flow along null geodesics, we have constructed a new gauge such that, in this gauge, the Einstein equation is well posed and such that the linearisation is Chrzanowski’s outgoing radiation gauge.

This is joint work with Lars Andersson, Thomas Backdahl, and Siyuan Ma.

10:30 am–11:30 am

Peter Hintz, MIT (virtual)

Title: Mode stability and shallow quasinormal modes of Kerr-de Sitter black holes

Abstract: The Kerr-de Sitter metric describes a rotating black hole with mass $m$ and specific angular momentum $a$ in a universe, such as our own, with cosmological constant $\Lambda>0$. I will explain a proof of mode stability for the scalar wave equation on Kerr-de Sitter spacetimes in the following setting: fixing $\Lambda$ and the ratio $|a/m|<1$ (related to the subextremality of the black hole in question), mode stability holds for sufficiently small black hole mass $m$. We also obtain estimates for the location of quasinormal modes (resonances) $\sigma$ in any fixed half space $\Im\sigma>-C$. Our results imply that solutions of the wave equation decay exponentially in time to constants, with an explicit exponential rate. The proof is based on careful uniform estimates for the spectral family in the singular limit $m\to 0$ in which, depending on the scaling, the Kerr-de Sitter spacetime limits to a Kerr or the de Sitter spacetime.

11:30 am–12:30 pm

Lars Andersson, Albert Einstein Institute, Germany (virtual)

Title: Gravitational instantons and special geometry

Abstract: Gravitational instantons are Ricci flat complete Riemannian 4-manifolds with at least quadratic curvature decay. In this talk, I will introduce some notions of special geometry, discuss known examples, and mention some open questions. The Chen-Teo gravitational instanton is an asymptotically flat, toric, Ricci flat family of metrics on $\mathrm{CP}^2 \setminus \mathrm{S}^1$, that provides a counterexample to the classical Euclidean Black Hole Uniqueness conjecture. I will sketch a proof that the Chen-Teo Instanton is Hermitian and non-Kähler. Thus, all known examples of gravitational instantons are Hermitian. This talks is based on joint work with Steffen Aksteiner, cf. https://arxiv.org/abs/2112.11863.

12:30 pm–1:30 pm

break

1:30 pm–2:30 pm

Martin Taylor, Imperial College London (virtual)

Title: The nonlinear stability of the Schwarzschild family of black holes

Abstract: I will present a theorem on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes. The proof employs a double null gauge, is expressed entirely in physical space, and utilises the analysis of Dafermos–Holzegel–Rodnianski on the linear stability of the Schwarzschild family. This is joint work with M. Dafermos, G. Holzegel and I. Rodnianski.

2:30 pm–3:30 pm

Po-Ning Chen, University of California, Riverside (virtual)

Title: Angular momentum in general relativity Abstract: The definition of angular momentum in general relativity has been a subtle issue since the 1960s, due to the ‘supertranslation ambiguity’. In this talk, we will discuss how the mathematical theory of quasilocal mass and angular momentum leads to a new definition of angular momentum at null infinity that is free of any supertranslation ambiguity.

This is based on joint work with Jordan Keller, Mu-Tao Wang, Ye-Kai Wang, and Shing-Tung Yau.

3:30 pm–4:00 pm

break

4:00 pm–5:00 pm

Dan Lee, Queens College (CUNY) (hybrid: in person & virtual)

Title: Stability of the positive mass theorem

Abstract: We will discuss the problem of stability for the rigidity part of the Riemannian positive mass theorem, focusing on recent work with Kazaras and Khuri, in which we proved that if one assumes a lower Ricci curvature bound, then stability holds with respect to pointed Gromov-Hausdorff convergence.

Tuesday, April 5, 2022

Time (ET)

Speaker

Title/Abstract

9:30 am–10:30 am

Xinliang An, National University of Singapore (virtual)

Title: Anisotropic dynamical horizons arising in gravitational collapse

Abstract: Black holes are predicted by Einstein’s theory of general relativity, and now we have ample observational evidence for their existence. However theoretically there are many unanswered questions about how black holes come into being and about the structures of their inner spacetime singularities. In this talk, we will present several results in these directions. First, in a joint work with Qing Han, with tools from scale-critical hyperbolic method and non-perturbative elliptic techniques, with anisotropic characteristic initial data we prove that: in the process of gravitational collapse, a smooth and spacelike apparent horizon (dynamical horizon) emerges from general (both isotropic and anisotropic) initial data. This result extends the 2008 Christodoulou’s monumental work and it connects to black hole thermodynamics along the apparent horizon. Second, in joint works with Dejan Gajic and Ruixiang Zhang, for the spherically symmetric Einstein-scalar field system, we derive precise blow-up rates for various geometric quantities along the inner spacelike singularities. These rates obey polynomial blow-up upper bounds; and when it is close to timelike infinity, these rates are not limited to discrete finite choices and they are related to the Price’s law along the event horizon. This indicates a new blow-up phenomenon, driven by a PDE mechanism, rather than an ODE mechanism. If time permits, some results on fluid dynamics will also be addressed.

10:30 am–11:30 am

Sergiu Klainerman, Princeton (virtual)

Title: Nonlinear stability of slowly rotating Kerr solutions

Abstract: I will talk about the status of the stability of Kerr conjecture in General Relativity based on recent results obtained in collaboration with Jeremie Szeftel and Elena Giorgi.

11:30 am–12:30 pm

Siyuan Ma, Sorbonne University (virtual)

Title: Sharp decay for Teukolsky master equation

Abstract: I will talk about joint work with L. Zhang on deriving the late time dynamics of the spin $s$ components that satisfy the Teukolsky master equation in Kerr spacetimes.

12:30 pm–1:30 pm

Break

1:30 pm–2:30 pm

Jonathan Luk, Stanford (virtual)

Title: A tale of two tails

Abstract: Motivated by the strong cosmic censorship conjecture, we introduce a general method for understanding the late-time tail for solutions to wave equations on asymptotically flat spacetimes in odd spatial dimensions. A particular consequence of the method is a re-proof of Price’s law-type results, which concern the sharp decay rate of the late-time tails on stationary spacetimes. Moreover, we show that the late-time tails are in general different from the stationary case in the presence of dynamical and/or nonlinear perturbations. This is a joint work with Sung-Jin Oh (Berkeley).

2:30 pm–3:30 pm

Gary Horowitz, University of California Santa Barbara (virtual)

Title: A new type of extremal black hole

Abstract: I describe a family of four-dimensional, asymptotically flat, charged black holes that develop (charged) scalar hair as one increases their charge at fixed mass. Surprisingly, the maximum charge for given mass is a nonsingular hairy black hole with a nondegenerate event horizon. Since the surface gravity is nonzero, if quantum matter is added, Hawking radiation does not appear to stop when this new extremal limit is reached. This raises the question of whether Hawking radiation will cause the black hole to turn into a naked singularity. I will argue that does not occur.

3:30 pm–4:00 pm

Break

4:00 pm–5:00 pm

Lydia Bieri, University of Michigan (virtual)

Title: Gravitational radiation in general spacetimes

Abstract: Studies of gravitational waves have been devoted mostly to sources such as binary black hole mergers or neutron star mergers, or generally sources that are stationary outside of a compact set. These systems are described by asymptotically-flat manifolds solving the Einstein equations with sufficiently fast decay of the gravitational field towards Minkowski spacetime far away from the source. Waves from such sources have been recorded by the LIGO/VIRGO collaboration since 2015. In this talk, I will present new results on gravitational radiation for sources that are not stationary outside of a compact set, but whose gravitational fields decay more slowly towards infinity. A panorama of new gravitational effects opens up when delving deeper into these more general spacetimes. In particular, whereas the former sources produce memory effects that are finite and of purely electric parity, the latter in addition generate memory of magnetic type, and both types grow. These new effects emerge naturally from the Einstein equations both in the Einstein vacuum case and for neutrino radiation. The latter results are important for sources with extended neutrino halos.

Wednesday, April 6, 2022

Time (ET)

Speaker

Title/Abstract

9:30 am–10:30 am

Gerhard Huisken, Mathematisches Forschungsinstitut Oberwolfach (virtual)

TBA

10:30 am–11:30 am

Carla Cederbaum, Universität Tübingen, Germany (virtual)

Title: Coordinates are messy

Abstract: Asymptotically Euclidean initial data sets $(M,g,K)$ are characterized by the existence of asymptotic coordinates in which the Riemannian metric $g$ and second fundamental form $K$ decay to the Euclidean metric $\delta$ and to $0$ suitably fast, respectively. Provided their matter densities satisfy suitable integrability conditions, they have well-defined (ADM-)energy, (ADM-)linear momentum, and (ADM-)mass. This was proven by Bartnik using harmonic coordinates. To study their (ADM-)angular momentum and (BORT-)center of mass, one usually assumes the existence of Regge—Teitelboim coordinates on the initial data set $(M,g,K)$ in question. We will give examples of asymptotically Euclidean initial data sets which do not possess any Regge—Teitelboim coordinates We will also show that harmonic coordinates can be used as a tool in checking whether a given asymptotically Euclidean initial data set possesses Regge—Teitelboim coordinates. This is joint work with Melanie Graf and Jan Metzger. We will also explain the consequences these findings have for the definition of the center of mass, relying on joint work with Nerz and with Sakovich.

11:30 am–12:30 pm

Stefanos Aretakis, University of Toronto (virtual)

Title: Observational signatures for extremal black holes

Abstract: We will present results regarding the asymptotics of scalar perturbations on black hole backgrounds. We will then derive observational signatures for extremal black holes that are based on global or localized measurements on null infinity. This is based on joint work with Gajic-Angelopoulos and ongoing work with Khanna-Sabharwal.

12:30 pm–1:30 pm

Break

1:30 pm–2:30 pm

Jared Speck, Vanderbilt University (virtual)

Title: The mathematical theory of shock waves in multi-dimensional relativistic and non-relativistic compressible Euler flow

Abstract: In the last two decades, there have been dramatic advances in the rigorous mathematical theory of shock waves in solutions to the relativistic Euler equations and their non-relativistic analog, the compressible Euler equations. A lot of the progress has relied on techniques that were developed to study Einstein’s equations. In this talk, I will provide an overview of the field and highlight some recent progress on problems without symmetry or irrotationality assumptions. I will focus on results that reveal various aspects of the structure of the maximal development of the data and the corresponding implications for the shock development problem, which is the problem of continuing the solution weakly after a shock. I will also describe various open problems, some of which are tied to the Einstein–Euler equations. Various aspects of this program are joint with L. Abbrescia, M. Disconzi, and J. Luk.

2:30 pm–3:30 pm

Lan-Hsuan Huang, University of Connecticut (hybrid: in person & virtual)

Title: Null perfect fluids, improvability of dominant energy scalar, and Bartnik mass minimizers

Abstract: We introduce the concept of improvability of the dominant energy scalar, and we derive strong consequences of non-improvability. In particular, we prove that a non-improvable initial data set without local symmetries must sit inside a null perfect fluid spacetime carrying a global Killing vector field. We also show that the dominant energy scalar is always almost improvable in a precise sense. Using these main results, we provide a characterization of Bartnik mass minimizing initial data sets which makes substantial progress toward Bartnik’s stationary conjecture.

Along the way we observe that in dimensions greater than eight there exist pp-wave counterexamples (without the optimal decay rate for asymptotically flatness) to the equality case of the spacetime positive mass theorem. As a consequence, we find counterexamples to Bartnik’s stationary and strict positivity conjectures in those dimensions. This talk is based on joint work with Dan A. Lee.

3:30 pm–4:00 pm

Break

4:00 pm–5:00 pm

Demetre Kazaras, Duke University (virtual)

Title: Comparison geometry for scalar curvature and spacetime harmonic functions

Abstract: Comparison theorems are the basis for our geometric understanding of Riemannian manifolds satisfying a given curvature condition. A remarkable example is the Gromov-Lawson toric band inequality, which bounds the distance between the two sides of a Riemannian torus-cross-interval with positive scalar curvature by a sharp constant inversely proportional to the scalar curvature’s minimum. We will give a new qualitative version of this and similar band-type inequalities in dimension 3 using the notion of spacetime harmonic functions, which recently played the lead role in our recent proof of the positive mass theorem. This is joint work with Sven Hirsch, Marcus Khuri, and Yiyue Zhang.

Thursday, April 7, 2022

Time (ET)

Speaker

Title/Abstract

9:30 am–10:30 am

Piotr Chrusciel, Universitat Wien (virtual)

Title: Maskit gluing and hyperbolic mass

Abstract: “Maskit gluing” is a gluing construction for asymptotically locally hyperbolic (ALH) manifolds with negative cosmological constant. I will present a formula for the mass of Maskit-glued ALH manifolds and describe how it can be used to construct general relativistic initial data with negative mass.

10:30 am–11:30 am

Greg Galloway, University of Miami (virtual)

Title: Initial data rigidity and applications

Abstract: We present a result from our work with Michael Eichmair and Abraão Mendes concerning initial data rigidity results (CMP, 2021), and look at some consequences. In a note with Piotr Chruściel (CQG 2021), we showed how to use this result, together with arguments from Chruściel and Delay’s proof of the their hyperbolic PMT result, to obtain a hyperbolic PMT result with boundary. This will also be discussed.

11:30 am–12:30 pm

Pengzi Miao, University of Miami (virtual)

Title: Some remarks on mass and quasi-local mass

Abstract: In the first part of this talk, I will describe how to detect the mass of asymptotically flat and asymptotically hyperbolic manifolds via large Riemannian polyhedra. In the second part, I will discuss an estimate of the Bartnik quasi-local mass and its geometric implications. This talk is based on several joint works with A. Piubello, and with H.C. Jang.

12:30 pm–1:30 pm

Break

1:30 pm–2:30 pm

Yakov Shlapentokh Rothman, Princeton (hybrid: in person & virtual)

Title: Self-Similarity and Naked Singularities for the Einstein Vacuum Equations

Abstract: We will start with an introduction to the problem of constructing naked singularities for the Einstein vacuum equations, and then explain our discovery of a fundamentally new type of self-similarity and show how this allows us to construct solutions corresponding to a naked singularity. This is joint work with Igor Rodnianski.

2:30 pm–3:30 pm

Marcelo Disconzi, Vanderbilt University (virtual)

Title: General-relativistic viscous fluids.

Abstract: The discovery of the quark-gluon plasma that forms in heavy-ion collision experiments provides a unique opportunity to study the properties of matter under extreme conditions, as the quark-gluon plasma is the hottest, smallest, and densest fluid known to humanity. Studying the quark-gluon plasma also provides a window into the earliest moments of the universe, since microseconds after the Big Bang the universe was filled with matter in the form of the quark-gluon plasma. For more than two decades, the community has intensely studied the quark-gluon plasma with the help of a rich interaction between experiments, theory, phenomenology, and numerical simulations. From these investigations, a coherent picture has emerged, indicating that the quark-gluon plasma behaves essentially like a relativistic liquid with viscosity. More recently, state-of-the-art numerical simulations strongly suggested that viscous and dissipative effects can also have non-negligible effects on gravitational waves produced by binary neutron star mergers. But despite the importance of viscous effects for the study of such systems, a robust and mathematically sound theory of relativistic fluids with viscosity is still lacking. This is due, in part, to difficulties to preserve causality upon the inclusion of viscous and dissipative effects into theories of relativistic fluids. In this talk, we will survey the history of the problem and report on a new approach to relativistic viscous fluids that addresses these issues.

3:30 pm–4:00 pm

Break

4:00 pm–5:00 pm

Maxime van de Moortel, Princeton (hybrid: in person & virtual)

Title: Black holes: the inside story of gravitational collapse

Abstract: What is inside a dynamical black hole? While the local region near time-like infinity is understood for various models, the global structure of the black hole interior has largely remained unexplored. These questions are deeply connected to the nature of singularities in General Relativity and celebrated problems such as Penrose’s Strong Cosmic Censorship Conjecture. I will present my recent resolution of these problems in spherical gravitational collapse, based on the discovery of a novel phenomenon: the breakdown of weak singularities and the dynamical formation of a strong singularity.

Friday, April 8, 2022

Time (ET)

Speaker

Title/Abstract

9:30 am–10:30 am

Ye-Kai Wang, National Cheng Kun University, Taiwan (virtual)

Title: Supertranslation invariance of angular momentum at null infinity in double null gauge

Abstract: This talk accompanies Po-Ning Chen’s talk on Monday with the results described in the double null gauge rather than Bondi-Sachs coordinates. Besides discussing how Chen-Wang-Yau angular momentum resolves the supertranslation ambiguity, we also review the definition of angular momentum defined by A. Rizzi. The talk is based on the joint work with Po-Ning Chen, Jordan Keller, Mu-Tao Wang, and Shing-Tung Yau.

10:30 am–11:30 am

Zoe Wyatt, King’s College London (virtual)

Title: Global Stability of Spacetimes with Supersymmetric Compactifications

Abstract: Spacetimes with compact directions which have special holonomy, such as Calabi-Yau spaces, play an important role in supergravity and string theory. In this talk I will discuss a recent work with Lars Andersson, Pieter Blue and Shing-Tung Yau, where we show the global, nonlinear stability a spacetime which is a cartesian product of a high dimensional Minkowski space with a compact Ricci flat internal space with special holonomy. This stability result is related to a conjecture of Penrose concerning the validity of string theory. Our proof uses the intersection of methods for quasilinear wave and Klein-Gordon equations, and so towards the end of the talk I will also comment more generally on coupled wave–Klein-Gordon equations.

11:30 am–12:30 pm

Elena Giorgi, Columbia University (hybrid: in person & virtual)

Title: The stability of charged black holes

Abstract: Black hole solutions in General Relativity are parametrized by their mass, spin and charge. In this talk, I will motivate why the charge of black holes adds interesting dynamics to solutions of the Einstein equation thanks to the interaction between gravitational and electromagnetic radiation. Such radiations are solutions of a system of coupled wave equations with a symmetric structure which allows to define a combined energy-momentum tensor for the system. Finally, I will show how this physical-space approach is resolutive in the most general case of Kerr-Newman black hole, where the interaction between the radiations prevents the separability in modes.

12:30 pm–1:30 pm

Break

1:30 pm–2:30 pm

Marcus Khuri, Stony Brook University (virtual)

Title: The mass-angular momentum inequality for multiple black holes Abstract: Consider a complete 3-dimensional initial data set for the Einstein equations which has multiple asymptotically flat or asymptotically cylindrical ends. If it is simply connected, axisymmetric, maximal, and satisfies the appropriate energy condition then the ADM mass of any of the asymptotically flat ends is bounded below by the square root of the total angular momentum. This generalizes previous work of Dain, Chrusciel-Li-Weinstein, and Schoen-Zhou which treated either the single black hole case or the multiple black hole case without an explicit lower bound. The proof relies on an analysis of the asymptotics of singular harmonic maps from R^3 \ \Gamma –>H^2 where \Gamma is a coordinate axis. This is joint work with Q. Han, G. Weinstein, and J. Xiong.

2:30 pm–3:30 pm

Martin Lesourd, Harvard (hybrid: in person & virtual)

Title: A Snippet on Mass and the Topology and Geometry of Positive Scalar Curvature

Abstract: I will talk about a small corner of the study of Positive Scalar Curvature (PSC) and questions which are most closely related to the Positive Mass Theorem. The classic questions are ”which topologies allow for PSC?” and ”what is the geometry of manifolds with PSC?”. This is based on joint work with Prof. S-T. Yau, Prof. D. A. Lee, and R. Unger.

3:30 pm–4:00 pm

Break

4:00 pm–5:00 pm

Georgios Moschidis, Princeton (virtual)

Title: Weak turbulence for the Einstein–scalar field system.

Abstract: In the presence of confinement, the Einstein field equations are expected to exhibit turbulent dynamics. In the presence of a negative cosmological constant, the AdS instability conjecture claims the existence of arbitrarily small perturbations to the initial data of Anti-de Sitter spacetime which, under evolution by the vacuum Einstein equations with reflecting boundary conditions at conformal infinity, lead to the formation of black holes after sufficiently long time. In this talk, I will present a rigorous proof of the AdS instability conjecture in the setting of the spherically symmetric Einstein-scalar field system. The construction of the unstable initial data will require carefully designing a family of initial configurations of localized matter beams and estimating the exchange of energy taking place between interacting beams over long periods of time, as well as estimating the decoherence rate of those beams.

For the full schedule, please see: https://cmsa.fas.harvard.edu/gr-program/

This conference will be held virtually on Zoom. Registration is required. Webinar Registration

A few talks will be held in hybrid formats, with talks given from the CMSA seminar room, G-10. Advanced registration for in-person components is required. In-Person Registration

For more information, please see https://cmsa.fas.harvard.edu/gr-program/

The Harvard CMSA will be hosting a conference on General Relativity from April 4-8, 2022.

This conference will be held virtually on Zoom. Registration is required. Webinar Registration A few talks will be held in hybrid formats, with talks given from the CMSA seminar room, G-10. Advanced registration for in-person components is required. In-Person Registration

For more information, please see: https://cmsa.fas.harvard.edu/gr-program/

In the last three decades, the problem of consciousness – how and why physical systems such as the brain have conscious experiences – has received increasing attention among neuroscientists, psychologists, and philosophers. Recently, a decidedly mathematical perspective has emerged as well, which is now called Mathematical Consciousness Science. In this talk, I will give an introduction and overview of Mathematical Consciousness Science for mathematicians, including a bottom-up introduction to the problem of consciousness and how it is amenable to mathematical tools and methods.

For information on how to join, please see: https://cmsa.fas.harvard.edu/seminars-and-colloquium/

Characterizing many-body entanglement is one of the most important problems in quantum physics. We present our studies on the steady state von Neumann entropy and its transition in Brownian SYK models. For unitary evolution, we show that the correlations between different replicas account for the Page curve at late time, and a permutation group structure emerges in the large-N calculation. In the presence of measurements, we find a transition of von Neumann entropy from volume-law to area-law by increasing the measurement rate. We show that a proper replica limit can be taken, which shows that the transition occurs at the point of replica symmetry breaking.

For information on how to join, please see: https://cmsa.fas.harvard.edu/seminars-and-colloquium/

Let C be an algebraic curve over a number field. Faltings’s theorem on rational points on subvarieties of abelian varieties implies that all algebraic points on C arise in algebraic families, with finitely many exceptions. These exceptions are known as isolated points. We study how isolated points behave under morphisms and then specialize to the case of modular curves. We show that isolated points on X_1(n) push down to isolated points on a modular curve whose level is bounded by a constant that depends only on the j-invariant of the isolated point. This is joint work with A. Bourdon, O. Ejder, Y. Liu, and F. Odumodu.

Consider the Gibbs distribution of the hardcore model over all independent sets of a given graph, where the probability density of each independent set J is proportional to lambda^|J| where lambda is a parameter. We study the single-site update Markov chain known as the Glauber dynamics for generating independent sets from this distribution. In each step, the dynamics picks a vertex uniformly at random and updates its status (inside or outside the independent set) conditional on the status of all other vertices. We prove optimal (nearly linear) mixing time bounds of the Glauber dynamics on bounded-degree graphs when lambda < lambda_c, beyond which it is known the dynamics can be exponentially slow. To establish our result, we utilize and improve the spectral independence approach of Anari, Liu, and Oveis Gharan (2020) and show optimal mixing time of the Glauber dynamics for spin systems when the maximum eigenvalues of associated influence matrices are bounded.

In this talk I’ll gently survey various roles mathematics (often, but not always, in the form of machine learning) plays in our information ecosystem. I’ll discuss the math behind YouTube’s recommendation algorithm and Facebook’s News Feed algorithm and the impact the choice of objective function has on what society sees and thinks. I’ll explain how graph theory is used to quantitatively study the spread of news and misinformation on social media, and also how it is used to detect bot accounts. And I’ll explain the math behind deepfake photos and videos and text generating AI. No prior knowledge of machine learning or data science will be assumed, and the math will be accessible to all undergraduates.

Gromov-Witten invariants of holomorphic symplectic 4-folds vanish and one can consider the corresponding reduced theory. In this talk, we will explain a definition of Gopakumar-Vafa type invariants for such a reduced theory. These invariants are conjectured to be integers and have alternative interpretations using sheaf theoretic moduli spaces. Our conjecture is proved for the product of two K3 surfaces, which naturally leads to a closed formula of Fujiki constants of Chern classes of tangent bundles of Hilbert schemes of points on K3 surfaces. On a very general holomorphic symplectic 4-folds of K3^[2] type, our conjecture provides a Yau-Zaslow type formula for the number of isolated genus 2 curves of minimal degree. Based on joint works with Georg Oberdieck and Yukinobu Toda.

For more information, please see: http://www.ims.cuhk.edu.hk/cgi-bin/SeminarAdmin/bin/Web

Pieter Blue, University of Edinburgh, UK (virtual)

Title: Linear stability of the Kerr spacetime in the outgoing radiation gauge

Abstract: This talk will discuss a new gauge condition (i.e. coordinate condition) for the Einstein equation, the linearisation of the Einstein equation in this gauge, and the decay of solutions to the linearised Einstein equation around Kerr black holes in this gauge. The stability of the family of Kerr black holes under the evolution generated by the Einstein equation is a long-standing problem in mathematical relativity. In 1972, Teukolsky discovered equations governing certain components of the linearised curvature that are invariant under linearised gague transformations. In 1975, Chrzanowski introduced the “outgoing radiation gauge”, a condition on the linearised metric that allows for the construction of the linearised metric from the linearised curvature. In 2019, we proved decay for the metric constructed using Chrzanowski’s outgoing radiation gauge. Recently, using a flow along null geodesics, we have constructed a new gauge such that, in this gauge, the Einstein equation is well posed and such that the linearisation is Chrzanowski’s outgoing radiation gauge.

This is joint work with Lars Andersson, Thomas Backdahl, and Siyuan Ma.

10:30 am–11:30 am

Peter Hintz, MIT (virtual)

Title: Mode stability and shallow quasinormal modes of Kerr-de Sitter black holes

Abstract: The Kerr-de Sitter metric describes a rotating black hole with mass $m$ and specific angular momentum $a$ in a universe, such as our own, with cosmological constant $\Lambda>0$. I will explain a proof of mode stability for the scalar wave equation on Kerr-de Sitter spacetimes in the following setting: fixing $\Lambda$ and the ratio $|a/m|<1$ (related to the subextremality of the black hole in question), mode stability holds for sufficiently small black hole mass $m$. We also obtain estimates for the location of quasinormal modes (resonances) $\sigma$ in any fixed half space $\Im\sigma>-C$. Our results imply that solutions of the wave equation decay exponentially in time to constants, with an explicit exponential rate. The proof is based on careful uniform estimates for the spectral family in the singular limit $m\to 0$ in which, depending on the scaling, the Kerr-de Sitter spacetime limits to a Kerr or the de Sitter spacetime.

11:30 am–12:30 pm

Lars Andersson, Albert Einstein Institute, Germany (virtual)

Title: Gravitational instantons and special geometry

Abstract: Gravitational instantons are Ricci flat complete Riemannian 4-manifolds with at least quadratic curvature decay. In this talk, I will introduce some notions of special geometry, discuss known examples, and mention some open questions. The Chen-Teo gravitational instanton is an asymptotically flat, toric, Ricci flat family of metrics on $\mathrm{CP}^2 \setminus \mathrm{S}^1$, that provides a counterexample to the classical Euclidean Black Hole Uniqueness conjecture. I will sketch a proof that the Chen-Teo Instanton is Hermitian and non-Kähler. Thus, all known examples of gravitational instantons are Hermitian. This talks is based on joint work with Steffen Aksteiner, cf. https://arxiv.org/abs/2112.11863.

12:30 pm–1:30 pm

break

1:30 pm–2:30 pm

Martin Taylor, Imperial College London (virtual)

Title: The nonlinear stability of the Schwarzschild family of black holes

Abstract: I will present a theorem on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes. The proof employs a double null gauge, is expressed entirely in physical space, and utilises the analysis of Dafermos–Holzegel–Rodnianski on the linear stability of the Schwarzschild family. This is joint work with M. Dafermos, G. Holzegel and I. Rodnianski.

2:30 pm–3:30 pm

Po-Ning Chen, University of California, Riverside (virtual)

Title: Angular momentum in general relativity Abstract: The definition of angular momentum in general relativity has been a subtle issue since the 1960s, due to the ‘supertranslation ambiguity’. In this talk, we will discuss how the mathematical theory of quasilocal mass and angular momentum leads to a new definition of angular momentum at null infinity that is free of any supertranslation ambiguity.

This is based on joint work with Jordan Keller, Mu-Tao Wang, Ye-Kai Wang, and Shing-Tung Yau.

3:30 pm–4:00 pm

break

4:00 pm–5:00 pm

Dan Lee, Queens College (CUNY) (hybrid: in person & virtual)

Title: Stability of the positive mass theorem

Abstract: We will discuss the problem of stability for the rigidity part of the Riemannian positive mass theorem, focusing on recent work with Kazaras and Khuri, in which we proved that if one assumes a lower Ricci curvature bound, then stability holds with respect to pointed Gromov-Hausdorff convergence.

Tuesday, April 5, 2022

Time (ET)

Speaker

Title/Abstract

9:30 am–10:30 am

Xinliang An, National University of Singapore (virtual)

Title: Anisotropic dynamical horizons arising in gravitational collapse

Abstract: Black holes are predicted by Einstein’s theory of general relativity, and now we have ample observational evidence for their existence. However theoretically there are many unanswered questions about how black holes come into being and about the structures of their inner spacetime singularities. In this talk, we will present several results in these directions. First, in a joint work with Qing Han, with tools from scale-critical hyperbolic method and non-perturbative elliptic techniques, with anisotropic characteristic initial data we prove that: in the process of gravitational collapse, a smooth and spacelike apparent horizon (dynamical horizon) emerges from general (both isotropic and anisotropic) initial data. This result extends the 2008 Christodoulou’s monumental work and it connects to black hole thermodynamics along the apparent horizon. Second, in joint works with Dejan Gajic and Ruixiang Zhang, for the spherically symmetric Einstein-scalar field system, we derive precise blow-up rates for various geometric quantities along the inner spacelike singularities. These rates obey polynomial blow-up upper bounds; and when it is close to timelike infinity, these rates are not limited to discrete finite choices and they are related to the Price’s law along the event horizon. This indicates a new blow-up phenomenon, driven by a PDE mechanism, rather than an ODE mechanism. If time permits, some results on fluid dynamics will also be addressed.

10:30 am–11:30 am

Sergiu Klainerman, Princeton (virtual)

Title: Nonlinear stability of slowly rotating Kerr solutions

Abstract: I will talk about the status of the stability of Kerr conjecture in General Relativity based on recent results obtained in collaboration with Jeremie Szeftel and Elena Giorgi.

11:30 am–12:30 pm

Siyuan Ma, Sorbonne University (virtual)

Title: Sharp decay for Teukolsky master equation

Abstract: I will talk about joint work with L. Zhang on deriving the late time dynamics of the spin $s$ components that satisfy the Teukolsky master equation in Kerr spacetimes.

12:30 pm–1:30 pm

Break

1:30 pm–2:30 pm

Jonathan Luk, Stanford (virtual)

Title: A tale of two tails

Abstract: Motivated by the strong cosmic censorship conjecture, we introduce a general method for understanding the late-time tail for solutions to wave equations on asymptotically flat spacetimes in odd spatial dimensions. A particular consequence of the method is a re-proof of Price’s law-type results, which concern the sharp decay rate of the late-time tails on stationary spacetimes. Moreover, we show that the late-time tails are in general different from the stationary case in the presence of dynamical and/or nonlinear perturbations. This is a joint work with Sung-Jin Oh (Berkeley).

2:30 pm–3:30 pm

Gary Horowitz, University of California Santa Barbara (virtual)

Title: A new type of extremal black hole

Abstract: I describe a family of four-dimensional, asymptotically flat, charged black holes that develop (charged) scalar hair as one increases their charge at fixed mass. Surprisingly, the maximum charge for given mass is a nonsingular hairy black hole with a nondegenerate event horizon. Since the surface gravity is nonzero, if quantum matter is added, Hawking radiation does not appear to stop when this new extremal limit is reached. This raises the question of whether Hawking radiation will cause the black hole to turn into a naked singularity. I will argue that does not occur.

3:30 pm–4:00 pm

Break

4:00 pm–5:00 pm

Lydia Bieri, University of Michigan (virtual)

Title: Gravitational radiation in general spacetimes

Abstract: Studies of gravitational waves have been devoted mostly to sources such as binary black hole mergers or neutron star mergers, or generally sources that are stationary outside of a compact set. These systems are described by asymptotically-flat manifolds solving the Einstein equations with sufficiently fast decay of the gravitational field towards Minkowski spacetime far away from the source. Waves from such sources have been recorded by the LIGO/VIRGO collaboration since 2015. In this talk, I will present new results on gravitational radiation for sources that are not stationary outside of a compact set, but whose gravitational fields decay more slowly towards infinity. A panorama of new gravitational effects opens up when delving deeper into these more general spacetimes. In particular, whereas the former sources produce memory effects that are finite and of purely electric parity, the latter in addition generate memory of magnetic type, and both types grow. These new effects emerge naturally from the Einstein equations both in the Einstein vacuum case and for neutrino radiation. The latter results are important for sources with extended neutrino halos.

Wednesday, April 6, 2022

Time (ET)

Speaker

Title/Abstract

9:30 am–10:30 am

Gerhard Huisken, Mathematisches Forschungsinstitut Oberwolfach (virtual)

TBA

10:30 am–11:30 am

Carla Cederbaum, Universität Tübingen, Germany (virtual)

Title: Coordinates are messy

Abstract: Asymptotically Euclidean initial data sets $(M,g,K)$ are characterized by the existence of asymptotic coordinates in which the Riemannian metric $g$ and second fundamental form $K$ decay to the Euclidean metric $\delta$ and to $0$ suitably fast, respectively. Provided their matter densities satisfy suitable integrability conditions, they have well-defined (ADM-)energy, (ADM-)linear momentum, and (ADM-)mass. This was proven by Bartnik using harmonic coordinates. To study their (ADM-)angular momentum and (BORT-)center of mass, one usually assumes the existence of Regge—Teitelboim coordinates on the initial data set $(M,g,K)$ in question. We will give examples of asymptotically Euclidean initial data sets which do not possess any Regge—Teitelboim coordinates We will also show that harmonic coordinates can be used as a tool in checking whether a given asymptotically Euclidean initial data set possesses Regge—Teitelboim coordinates. This is joint work with Melanie Graf and Jan Metzger. We will also explain the consequences these findings have for the definition of the center of mass, relying on joint work with Nerz and with Sakovich.

11:30 am–12:30 pm

Stefanos Aretakis, University of Toronto (virtual)

Title: Observational signatures for extremal black holes

Abstract: We will present results regarding the asymptotics of scalar perturbations on black hole backgrounds. We will then derive observational signatures for extremal black holes that are based on global or localized measurements on null infinity. This is based on joint work with Gajic-Angelopoulos and ongoing work with Khanna-Sabharwal.

12:30 pm–1:30 pm

Break

1:30 pm–2:30 pm

Jared Speck, Vanderbilt University (virtual)

Title: The mathematical theory of shock waves in multi-dimensional relativistic and non-relativistic compressible Euler flow

Abstract: In the last two decades, there have been dramatic advances in the rigorous mathematical theory of shock waves in solutions to the relativistic Euler equations and their non-relativistic analog, the compressible Euler equations. A lot of the progress has relied on techniques that were developed to study Einstein’s equations. In this talk, I will provide an overview of the field and highlight some recent progress on problems without symmetry or irrotationality assumptions. I will focus on results that reveal various aspects of the structure of the maximal development of the data and the corresponding implications for the shock development problem, which is the problem of continuing the solution weakly after a shock. I will also describe various open problems, some of which are tied to the Einstein–Euler equations. Various aspects of this program are joint with L. Abbrescia, M. Disconzi, and J. Luk.

2:30 pm–3:30 pm

Lan-Hsuan Huang, University of Connecticut (hybrid: in person & virtual)

Title: Null perfect fluids, improvability of dominant energy scalar, and Bartnik mass minimizers

Abstract: We introduce the concept of improvability of the dominant energy scalar, and we derive strong consequences of non-improvability. In particular, we prove that a non-improvable initial data set without local symmetries must sit inside a null perfect fluid spacetime carrying a global Killing vector field. We also show that the dominant energy scalar is always almost improvable in a precise sense. Using these main results, we provide a characterization of Bartnik mass minimizing initial data sets which makes substantial progress toward Bartnik’s stationary conjecture.

Along the way we observe that in dimensions greater than eight there exist pp-wave counterexamples (without the optimal decay rate for asymptotically flatness) to the equality case of the spacetime positive mass theorem. As a consequence, we find counterexamples to Bartnik’s stationary and strict positivity conjectures in those dimensions. This talk is based on joint work with Dan A. Lee.

3:30 pm–4:00 pm

Break

4:00 pm–5:00 pm

Demetre Kazaras, Duke University (virtual)

Title: Comparison geometry for scalar curvature and spacetime harmonic functions

Abstract: Comparison theorems are the basis for our geometric understanding of Riemannian manifolds satisfying a given curvature condition. A remarkable example is the Gromov-Lawson toric band inequality, which bounds the distance between the two sides of a Riemannian torus-cross-interval with positive scalar curvature by a sharp constant inversely proportional to the scalar curvature’s minimum. We will give a new qualitative version of this and similar band-type inequalities in dimension 3 using the notion of spacetime harmonic functions, which recently played the lead role in our recent proof of the positive mass theorem. This is joint work with Sven Hirsch, Marcus Khuri, and Yiyue Zhang.

Thursday, April 7, 2022

Time (ET)

Speaker

Title/Abstract

9:30 am–10:30 am

Piotr Chrusciel, Universitat Wien (virtual)

Title: Maskit gluing and hyperbolic mass

Abstract: “Maskit gluing” is a gluing construction for asymptotically locally hyperbolic (ALH) manifolds with negative cosmological constant. I will present a formula for the mass of Maskit-glued ALH manifolds and describe how it can be used to construct general relativistic initial data with negative mass.

10:30 am–11:30 am

Greg Galloway, University of Miami (virtual)

Title: Initial data rigidity and applications

Abstract: We present a result from our work with Michael Eichmair and Abraão Mendes concerning initial data rigidity results (CMP, 2021), and look at some consequences. In a note with Piotr Chruściel (CQG 2021), we showed how to use this result, together with arguments from Chruściel and Delay’s proof of the their hyperbolic PMT result, to obtain a hyperbolic PMT result with boundary. This will also be discussed.

11:30 am–12:30 pm

Pengzi Miao, University of Miami (virtual)

Title: Some remarks on mass and quasi-local mass

Abstract: In the first part of this talk, I will describe how to detect the mass of asymptotically flat and asymptotically hyperbolic manifolds via large Riemannian polyhedra. In the second part, I will discuss an estimate of the Bartnik quasi-local mass and its geometric implications. This talk is based on several joint works with A. Piubello, and with H.C. Jang.

12:30 pm–1:30 pm

Break

1:30 pm–2:30 pm

Yakov Shlapentokh Rothman, Princeton (hybrid: in person & virtual)

Title: Self-Similarity and Naked Singularities for the Einstein Vacuum Equations

Abstract: We will start with an introduction to the problem of constructing naked singularities for the Einstein vacuum equations, and then explain our discovery of a fundamentally new type of self-similarity and show how this allows us to construct solutions corresponding to a naked singularity. This is joint work with Igor Rodnianski.

2:30 pm–3:30 pm

Marcelo Disconzi, Vanderbilt University (virtual)

Title: General-relativistic viscous fluids.

Abstract: The discovery of the quark-gluon plasma that forms in heavy-ion collision experiments provides a unique opportunity to study the properties of matter under extreme conditions, as the quark-gluon plasma is the hottest, smallest, and densest fluid known to humanity. Studying the quark-gluon plasma also provides a window into the earliest moments of the universe, since microseconds after the Big Bang the universe was filled with matter in the form of the quark-gluon plasma. For more than two decades, the community has intensely studied the quark-gluon plasma with the help of a rich interaction between experiments, theory, phenomenology, and numerical simulations. From these investigations, a coherent picture has emerged, indicating that the quark-gluon plasma behaves essentially like a relativistic liquid with viscosity. More recently, state-of-the-art numerical simulations strongly suggested that viscous and dissipative effects can also have non-negligible effects on gravitational waves produced by binary neutron star mergers. But despite the importance of viscous effects for the study of such systems, a robust and mathematically sound theory of relativistic fluids with viscosity is still lacking. This is due, in part, to difficulties to preserve causality upon the inclusion of viscous and dissipative effects into theories of relativistic fluids. In this talk, we will survey the history of the problem and report on a new approach to relativistic viscous fluids that addresses these issues.

3:30 pm–4:00 pm

Break

4:00 pm–5:00 pm

Maxime van de Moortel, Princeton (hybrid: in person & virtual)

Title: Black holes: the inside story of gravitational collapse

Abstract: What is inside a dynamical black hole? While the local region near time-like infinity is understood for various models, the global structure of the black hole interior has largely remained unexplored. These questions are deeply connected to the nature of singularities in General Relativity and celebrated problems such as Penrose’s Strong Cosmic Censorship Conjecture. I will present my recent resolution of these problems in spherical gravitational collapse, based on the discovery of a novel phenomenon: the breakdown of weak singularities and the dynamical formation of a strong singularity.

Friday, April 8, 2022

Time (ET)

Speaker

Title/Abstract

9:30 am–10:30 am

Ye-Kai Wang, National Cheng Kun University, Taiwan (virtual)

Title: Supertranslation invariance of angular momentum at null infinity in double null gauge

Abstract: This talk accompanies Po-Ning Chen’s talk on Monday with the results described in the double null gauge rather than Bondi-Sachs coordinates. Besides discussing how Chen-Wang-Yau angular momentum resolves the supertranslation ambiguity, we also review the definition of angular momentum defined by A. Rizzi. The talk is based on the joint work with Po-Ning Chen, Jordan Keller, Mu-Tao Wang, and Shing-Tung Yau.

10:30 am–11:30 am

Zoe Wyatt, King’s College London (virtual)

Title: Global Stability of Spacetimes with Supersymmetric Compactifications

Abstract: Spacetimes with compact directions which have special holonomy, such as Calabi-Yau spaces, play an important role in supergravity and string theory. In this talk I will discuss a recent work with Lars Andersson, Pieter Blue and Shing-Tung Yau, where we show the global, nonlinear stability a spacetime which is a cartesian product of a high dimensional Minkowski space with a compact Ricci flat internal space with special holonomy. This stability result is related to a conjecture of Penrose concerning the validity of string theory. Our proof uses the intersection of methods for quasilinear wave and Klein-Gordon equations, and so towards the end of the talk I will also comment more generally on coupled wave–Klein-Gordon equations.

11:30 am–12:30 pm

Elena Giorgi, Columbia University (hybrid: in person & virtual)

Title: The stability of charged black holes

Abstract: Black hole solutions in General Relativity are parametrized by their mass, spin and charge. In this talk, I will motivate why the charge of black holes adds interesting dynamics to solutions of the Einstein equation thanks to the interaction between gravitational and electromagnetic radiation. Such radiations are solutions of a system of coupled wave equations with a symmetric structure which allows to define a combined energy-momentum tensor for the system. Finally, I will show how this physical-space approach is resolutive in the most general case of Kerr-Newman black hole, where the interaction between the radiations prevents the separability in modes.

12:30 pm–1:30 pm

Break

1:30 pm–2:30 pm

Marcus Khuri, Stony Brook University (virtual)

Title: The mass-angular momentum inequality for multiple black holes Abstract: Consider a complete 3-dimensional initial data set for the Einstein equations which has multiple asymptotically flat or asymptotically cylindrical ends. If it is simply connected, axisymmetric, maximal, and satisfies the appropriate energy condition then the ADM mass of any of the asymptotically flat ends is bounded below by the square root of the total angular momentum. This generalizes previous work of Dain, Chrusciel-Li-Weinstein, and Schoen-Zhou which treated either the single black hole case or the multiple black hole case without an explicit lower bound. The proof relies on an analysis of the asymptotics of singular harmonic maps from R^3 \ \Gamma –>H^2 where \Gamma is a coordinate axis. This is joint work with Q. Han, G. Weinstein, and J. Xiong.

2:30 pm–3:30 pm

Martin Lesourd, Harvard (hybrid: in person & virtual)

Title: A Snippet on Mass and the Topology and Geometry of Positive Scalar Curvature

Abstract: I will talk about a small corner of the study of Positive Scalar Curvature (PSC) and questions which are most closely related to the Positive Mass Theorem. The classic questions are ”which topologies allow for PSC?” and ”what is the geometry of manifolds with PSC?”. This is based on joint work with Prof. S-T. Yau, Prof. D. A. Lee, and R. Unger.

3:30 pm–4:00 pm

Break

4:00 pm–5:00 pm

Georgios Moschidis, Princeton (virtual)

Title: Weak turbulence for the Einstein–scalar field system.

Abstract: In the presence of confinement, the Einstein field equations are expected to exhibit turbulent dynamics. In the presence of a negative cosmological constant, the AdS instability conjecture claims the existence of arbitrarily small perturbations to the initial data of Anti-de Sitter spacetime which, under evolution by the vacuum Einstein equations with reflecting boundary conditions at conformal infinity, lead to the formation of black holes after sufficiently long time. In this talk, I will present a rigorous proof of the AdS instability conjecture in the setting of the spherically symmetric Einstein-scalar field system. The construction of the unstable initial data will require carefully designing a family of initial configurations of localized matter beams and estimating the exchange of energy taking place between interacting beams over long periods of time, as well as estimating the decoherence rate of those beams.

For the full schedule, please see: https://cmsa.fas.harvard.edu/gr-program/

This conference will be held virtually on Zoom. Registration is required. Webinar Registration

A few talks will be held in hybrid formats, with talks given from the CMSA seminar room, G-10. Advanced registration for in-person components is required. In-Person Registration

For more information, please see https://cmsa.fas.harvard.edu/gr-program/

Bott periodicity relates vector bundles on a topological space X to vector bundles on X “times a sphere”. I’m not a topologist, so I will try to explain an algebraic or geometric incarnation, in terms of vector bundles on the Riemann sphere. I will attempt to make the talk introductory, and (for the most part) accessible to those in all fields, at the expense of speaking informally and not getting far. This relates to recent work of Hannah Larson, as well as joint work with (separately) Larson and Jim Bryan.

For information on how to join, please see: https://cmsa.fas.harvard.edu/seminars-and-colloquium/

It follows rather directly from Gödel that mathematical progress per mathematician should diminish with time. The real problem is what relation there is between human mathematics and the set of consequences of the ZFC axioms.

Exactly solvable spin models such as toric codes and X-cube model have heightened our understanding of spin liquids and topological matter in two and three dimensions. Their exact solvability, it turns out, is rooted in the existence of commuting generators in their parent lattice gauge theory (LGT). We can understand the toric codes as Higgsed descendants of the rank-1 U(1) LGT in two and three dimensions, and the X-cube model as that of rank-2 U(1) LGT in three dimensions. Furthermore, the transformation properties of the gauge fields in the respective LGT is responsible for, and nearly determines the structure of the effective field theory (EFT) of the accompanying matter fields. We show how to construct the EFT of e and m particles in the toric codes and of fractons and lineons in the X-cube model by following such an idea. Recently we proposed some stabilizer Hamiltonians termed rank-2 toric code (R2TC) and F3 model (3D). We will explain what they are, and construct their EFTs using the gauge principle as guidance. The resulting field theory of the matter fields are usually highly interacting and exhibit unusual conservation laws. Especially for the R2TC, we demonstrate the existence of what we call the “dipolar braiding statistics” and outline the accompanying field theory which differs from the usual BF field theory of anyon braiding.

For information on how to join, please see: https://cmsa.fas.harvard.edu/seminars-and-colloquium/

The Harvard CMSA will be hosting a conference on General Relativity from April 4-8, 2022.

This conference will be held virtually on Zoom. Registration is required. Webinar Registration A few talks will be held in hybrid formats, with talks given from the CMSA seminar room, G-10. Advanced registration for in-person components is required. In-Person Registration

For more information, please see: https://cmsa.fas.harvard.edu/gr-program/

Berkovich spaces over Z may be seen as fibrations containing complex analytic spaces as well as p-adic analytic spaces, for every prime number p. We will give an introduction to those spaces and explain how they may be used in an arithmetic context to prove height inequalities. As an application, following a strategy by DeMarco-Krieger-Ye, we will give a proof of a conjecture of Bogomolov-Fu-Tschinkel on uniform bounds on the number of common images on P^1 of torsion points of two elliptic curves.

The morphogenesis of branched tissues has been a subject of long-standing debate. Although much is known about the molecular pathways that control cell fate decisions, it remains unclear how macroscopic features of branched organs, including their size, network topology and spatial pattern are encoded. Based on large-scale reconstructions of the mouse mammary gland and kidney, we begin by showing that statistical features of the developing branched epithelium can be explained quantitatively by a local self-organizing principle based on a branching and annihilating random walk (BARW). In this model, renewing tip-localized progenitors drive a serial process of ductal elongation and stochastic tip bifurcation that terminates when active tips encounter maturing ducts. Then, based on reconstructions of the developing mouse salivary gland, we propose a generalisation of BARW model in which tips arrested through steric interaction with proximate ducts reactivate their branching programme as constraints become alleviated through the expansion of the underlying mesenchyme. This inflationary branching-arresting random walk model offers a more general paradigm for branching morphogenesis when the ductal epithelium grows cooperatively with the tissue into which it expands.

For information on how to join, please see: https://cmsa.fas.harvard.edu/seminars-and-colloquium/

Pieter Blue, University of Edinburgh, UK (virtual)

Title: Linear stability of the Kerr spacetime in the outgoing radiation gauge

Abstract: This talk will discuss a new gauge condition (i.e. coordinate condition) for the Einstein equation, the linearisation of the Einstein equation in this gauge, and the decay of solutions to the linearised Einstein equation around Kerr black holes in this gauge. The stability of the family of Kerr black holes under the evolution generated by the Einstein equation is a long-standing problem in mathematical relativity. In 1972, Teukolsky discovered equations governing certain components of the linearised curvature that are invariant under linearised gague transformations. In 1975, Chrzanowski introduced the “outgoing radiation gauge”, a condition on the linearised metric that allows for the construction of the linearised metric from the linearised curvature. In 2019, we proved decay for the metric constructed using Chrzanowski’s outgoing radiation gauge. Recently, using a flow along null geodesics, we have constructed a new gauge such that, in this gauge, the Einstein equation is well posed and such that the linearisation is Chrzanowski’s outgoing radiation gauge.

This is joint work with Lars Andersson, Thomas Backdahl, and Siyuan Ma.

10:30 am–11:30 am

Peter Hintz, MIT (virtual)

Title: Mode stability and shallow quasinormal modes of Kerr-de Sitter black holes

Abstract: The Kerr-de Sitter metric describes a rotating black hole with mass $m$ and specific angular momentum $a$ in a universe, such as our own, with cosmological constant $\Lambda>0$. I will explain a proof of mode stability for the scalar wave equation on Kerr-de Sitter spacetimes in the following setting: fixing $\Lambda$ and the ratio $|a/m|<1$ (related to the subextremality of the black hole in question), mode stability holds for sufficiently small black hole mass $m$. We also obtain estimates for the location of quasinormal modes (resonances) $\sigma$ in any fixed half space $\Im\sigma>-C$. Our results imply that solutions of the wave equation decay exponentially in time to constants, with an explicit exponential rate. The proof is based on careful uniform estimates for the spectral family in the singular limit $m\to 0$ in which, depending on the scaling, the Kerr-de Sitter spacetime limits to a Kerr or the de Sitter spacetime.

11:30 am–12:30 pm

Lars Andersson, Albert Einstein Institute, Germany (virtual)

Title: Gravitational instantons and special geometry

Abstract: Gravitational instantons are Ricci flat complete Riemannian 4-manifolds with at least quadratic curvature decay. In this talk, I will introduce some notions of special geometry, discuss known examples, and mention some open questions. The Chen-Teo gravitational instanton is an asymptotically flat, toric, Ricci flat family of metrics on $\mathrm{CP}^2 \setminus \mathrm{S}^1$, that provides a counterexample to the classical Euclidean Black Hole Uniqueness conjecture. I will sketch a proof that the Chen-Teo Instanton is Hermitian and non-Kähler. Thus, all known examples of gravitational instantons are Hermitian. This talks is based on joint work with Steffen Aksteiner, cf. https://arxiv.org/abs/2112.11863.

12:30 pm–1:30 pm

break

1:30 pm–2:30 pm

Martin Taylor, Imperial College London (virtual)

Title: The nonlinear stability of the Schwarzschild family of black holes

Abstract: I will present a theorem on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes. The proof employs a double null gauge, is expressed entirely in physical space, and utilises the analysis of Dafermos–Holzegel–Rodnianski on the linear stability of the Schwarzschild family. This is joint work with M. Dafermos, G. Holzegel and I. Rodnianski.

2:30 pm–3:30 pm

Po-Ning Chen, University of California, Riverside (virtual)

Title: Angular momentum in general relativity Abstract: The definition of angular momentum in general relativity has been a subtle issue since the 1960s, due to the ‘supertranslation ambiguity’. In this talk, we will discuss how the mathematical theory of quasilocal mass and angular momentum leads to a new definition of angular momentum at null infinity that is free of any supertranslation ambiguity.

This is based on joint work with Jordan Keller, Mu-Tao Wang, Ye-Kai Wang, and Shing-Tung Yau.

3:30 pm–4:00 pm

break

4:00 pm–5:00 pm

Dan Lee, Queens College (CUNY) (hybrid: in person & virtual)

Title: Stability of the positive mass theorem

Abstract: We will discuss the problem of stability for the rigidity part of the Riemannian positive mass theorem, focusing on recent work with Kazaras and Khuri, in which we proved that if one assumes a lower Ricci curvature bound, then stability holds with respect to pointed Gromov-Hausdorff convergence.

Tuesday, April 5, 2022

Time (ET)

Speaker

Title/Abstract

9:30 am–10:30 am

Xinliang An, National University of Singapore (virtual)

Title: Anisotropic dynamical horizons arising in gravitational collapse

Abstract: Black holes are predicted by Einstein’s theory of general relativity, and now we have ample observational evidence for their existence. However theoretically there are many unanswered questions about how black holes come into being and about the structures of their inner spacetime singularities. In this talk, we will present several results in these directions. First, in a joint work with Qing Han, with tools from scale-critical hyperbolic method and non-perturbative elliptic techniques, with anisotropic characteristic initial data we prove that: in the process of gravitational collapse, a smooth and spacelike apparent horizon (dynamical horizon) emerges from general (both isotropic and anisotropic) initial data. This result extends the 2008 Christodoulou’s monumental work and it connects to black hole thermodynamics along the apparent horizon. Second, in joint works with Dejan Gajic and Ruixiang Zhang, for the spherically symmetric Einstein-scalar field system, we derive precise blow-up rates for various geometric quantities along the inner spacelike singularities. These rates obey polynomial blow-up upper bounds; and when it is close to timelike infinity, these rates are not limited to discrete finite choices and they are related to the Price’s law along the event horizon. This indicates a new blow-up phenomenon, driven by a PDE mechanism, rather than an ODE mechanism. If time permits, some results on fluid dynamics will also be addressed.

10:30 am–11:30 am

Sergiu Klainerman, Princeton (virtual)

Title: Nonlinear stability of slowly rotating Kerr solutions

Abstract: I will talk about the status of the stability of Kerr conjecture in General Relativity based on recent results obtained in collaboration with Jeremie Szeftel and Elena Giorgi.

11:30 am–12:30 pm

Siyuan Ma, Sorbonne University (virtual)

Title: Sharp decay for Teukolsky master equation

Abstract: I will talk about joint work with L. Zhang on deriving the late time dynamics of the spin $s$ components that satisfy the Teukolsky master equation in Kerr spacetimes.

12:30 pm–1:30 pm

Break

1:30 pm–2:30 pm

Jonathan Luk, Stanford (virtual)

Title: A tale of two tails

Abstract: Motivated by the strong cosmic censorship conjecture, we introduce a general method for understanding the late-time tail for solutions to wave equations on asymptotically flat spacetimes in odd spatial dimensions. A particular consequence of the method is a re-proof of Price’s law-type results, which concern the sharp decay rate of the late-time tails on stationary spacetimes. Moreover, we show that the late-time tails are in general different from the stationary case in the presence of dynamical and/or nonlinear perturbations. This is a joint work with Sung-Jin Oh (Berkeley).

2:30 pm–3:30 pm

Gary Horowitz, University of California Santa Barbara (virtual)

Title: A new type of extremal black hole

Abstract: I describe a family of four-dimensional, asymptotically flat, charged black holes that develop (charged) scalar hair as one increases their charge at fixed mass. Surprisingly, the maximum charge for given mass is a nonsingular hairy black hole with a nondegenerate event horizon. Since the surface gravity is nonzero, if quantum matter is added, Hawking radiation does not appear to stop when this new extremal limit is reached. This raises the question of whether Hawking radiation will cause the black hole to turn into a naked singularity. I will argue that does not occur.

3:30 pm–4:00 pm

Break

4:00 pm–5:00 pm

Lydia Bieri, University of Michigan (virtual)

Title: Gravitational radiation in general spacetimes

Abstract: Studies of gravitational waves have been devoted mostly to sources such as binary black hole mergers or neutron star mergers, or generally sources that are stationary outside of a compact set. These systems are described by asymptotically-flat manifolds solving the Einstein equations with sufficiently fast decay of the gravitational field towards Minkowski spacetime far away from the source. Waves from such sources have been recorded by the LIGO/VIRGO collaboration since 2015. In this talk, I will present new results on gravitational radiation for sources that are not stationary outside of a compact set, but whose gravitational fields decay more slowly towards infinity. A panorama of new gravitational effects opens up when delving deeper into these more general spacetimes. In particular, whereas the former sources produce memory effects that are finite and of purely electric parity, the latter in addition generate memory of magnetic type, and both types grow. These new effects emerge naturally from the Einstein equations both in the Einstein vacuum case and for neutrino radiation. The latter results are important for sources with extended neutrino halos.

Wednesday, April 6, 2022

Time (ET)

Speaker

Title/Abstract

9:30 am–10:30 am

Gerhard Huisken, Mathematisches Forschungsinstitut Oberwolfach (virtual)

TBA

10:30 am–11:30 am

Carla Cederbaum, Universität Tübingen, Germany (virtual)

Title: Coordinates are messy

Abstract: Asymptotically Euclidean initial data sets $(M,g,K)$ are characterized by the existence of asymptotic coordinates in which the Riemannian metric $g$ and second fundamental form $K$ decay to the Euclidean metric $\delta$ and to $0$ suitably fast, respectively. Provided their matter densities satisfy suitable integrability conditions, they have well-defined (ADM-)energy, (ADM-)linear momentum, and (ADM-)mass. This was proven by Bartnik using harmonic coordinates. To study their (ADM-)angular momentum and (BORT-)center of mass, one usually assumes the existence of Regge—Teitelboim coordinates on the initial data set $(M,g,K)$ in question. We will give examples of asymptotically Euclidean initial data sets which do not possess any Regge—Teitelboim coordinates We will also show that harmonic coordinates can be used as a tool in checking whether a given asymptotically Euclidean initial data set possesses Regge—Teitelboim coordinates. This is joint work with Melanie Graf and Jan Metzger. We will also explain the consequences these findings have for the definition of the center of mass, relying on joint work with Nerz and with Sakovich.

11:30 am–12:30 pm

Stefanos Aretakis, University of Toronto (virtual)

Title: Observational signatures for extremal black holes

Abstract: We will present results regarding the asymptotics of scalar perturbations on black hole backgrounds. We will then derive observational signatures for extremal black holes that are based on global or localized measurements on null infinity. This is based on joint work with Gajic-Angelopoulos and ongoing work with Khanna-Sabharwal.

12:30 pm–1:30 pm

Break

1:30 pm–2:30 pm

Jared Speck, Vanderbilt University (virtual)

Title: The mathematical theory of shock waves in multi-dimensional relativistic and non-relativistic compressible Euler flow

Abstract: In the last two decades, there have been dramatic advances in the rigorous mathematical theory of shock waves in solutions to the relativistic Euler equations and their non-relativistic analog, the compressible Euler equations. A lot of the progress has relied on techniques that were developed to study Einstein’s equations. In this talk, I will provide an overview of the field and highlight some recent progress on problems without symmetry or irrotationality assumptions. I will focus on results that reveal various aspects of the structure of the maximal development of the data and the corresponding implications for the shock development problem, which is the problem of continuing the solution weakly after a shock. I will also describe various open problems, some of which are tied to the Einstein–Euler equations. Various aspects of this program are joint with L. Abbrescia, M. Disconzi, and J. Luk.

2:30 pm–3:30 pm

Lan-Hsuan Huang, University of Connecticut (hybrid: in person & virtual)

Title: Null perfect fluids, improvability of dominant energy scalar, and Bartnik mass minimizers

Abstract: We introduce the concept of improvability of the dominant energy scalar, and we derive strong consequences of non-improvability. In particular, we prove that a non-improvable initial data set without local symmetries must sit inside a null perfect fluid spacetime carrying a global Killing vector field. We also show that the dominant energy scalar is always almost improvable in a precise sense. Using these main results, we provide a characterization of Bartnik mass minimizing initial data sets which makes substantial progress toward Bartnik’s stationary conjecture.

Along the way we observe that in dimensions greater than eight there exist pp-wave counterexamples (without the optimal decay rate for asymptotically flatness) to the equality case of the spacetime positive mass theorem. As a consequence, we find counterexamples to Bartnik’s stationary and strict positivity conjectures in those dimensions. This talk is based on joint work with Dan A. Lee.

3:30 pm–4:00 pm

Break

4:00 pm–5:00 pm

Demetre Kazaras, Duke University (virtual)

Title: Comparison geometry for scalar curvature and spacetime harmonic functions

Abstract: Comparison theorems are the basis for our geometric understanding of Riemannian manifolds satisfying a given curvature condition. A remarkable example is the Gromov-Lawson toric band inequality, which bounds the distance between the two sides of a Riemannian torus-cross-interval with positive scalar curvature by a sharp constant inversely proportional to the scalar curvature’s minimum. We will give a new qualitative version of this and similar band-type inequalities in dimension 3 using the notion of spacetime harmonic functions, which recently played the lead role in our recent proof of the positive mass theorem. This is joint work with Sven Hirsch, Marcus Khuri, and Yiyue Zhang.

Thursday, April 7, 2022

Time (ET)

Speaker

Title/Abstract

9:30 am–10:30 am

Piotr Chrusciel, Universitat Wien (virtual)

Title: Maskit gluing and hyperbolic mass

Abstract: “Maskit gluing” is a gluing construction for asymptotically locally hyperbolic (ALH) manifolds with negative cosmological constant. I will present a formula for the mass of Maskit-glued ALH manifolds and describe how it can be used to construct general relativistic initial data with negative mass.

10:30 am–11:30 am

Greg Galloway, University of Miami (virtual)

Title: Initial data rigidity and applications

Abstract: We present a result from our work with Michael Eichmair and Abraão Mendes concerning initial data rigidity results (CMP, 2021), and look at some consequences. In a note with Piotr Chruściel (CQG 2021), we showed how to use this result, together with arguments from Chruściel and Delay’s proof of the their hyperbolic PMT result, to obtain a hyperbolic PMT result with boundary. This will also be discussed.

11:30 am–12:30 pm

Pengzi Miao, University of Miami (virtual)

Title: Some remarks on mass and quasi-local mass

Abstract: In the first part of this talk, I will describe how to detect the mass of asymptotically flat and asymptotically hyperbolic manifolds via large Riemannian polyhedra. In the second part, I will discuss an estimate of the Bartnik quasi-local mass and its geometric implications. This talk is based on several joint works with A. Piubello, and with H.C. Jang.

12:30 pm–1:30 pm

Break

1:30 pm–2:30 pm

Yakov Shlapentokh Rothman, Princeton (hybrid: in person & virtual)

Title: Self-Similarity and Naked Singularities for the Einstein Vacuum Equations

Abstract: We will start with an introduction to the problem of constructing naked singularities for the Einstein vacuum equations, and then explain our discovery of a fundamentally new type of self-similarity and show how this allows us to construct solutions corresponding to a naked singularity. This is joint work with Igor Rodnianski.

2:30 pm–3:30 pm

Marcelo Disconzi, Vanderbilt University (virtual)

Title: General-relativistic viscous fluids.

Abstract: The discovery of the quark-gluon plasma that forms in heavy-ion collision experiments provides a unique opportunity to study the properties of matter under extreme conditions, as the quark-gluon plasma is the hottest, smallest, and densest fluid known to humanity. Studying the quark-gluon plasma also provides a window into the earliest moments of the universe, since microseconds after the Big Bang the universe was filled with matter in the form of the quark-gluon plasma. For more than two decades, the community has intensely studied the quark-gluon plasma with the help of a rich interaction between experiments, theory, phenomenology, and numerical simulations. From these investigations, a coherent picture has emerged, indicating that the quark-gluon plasma behaves essentially like a relativistic liquid with viscosity. More recently, state-of-the-art numerical simulations strongly suggested that viscous and dissipative effects can also have non-negligible effects on gravitational waves produced by binary neutron star mergers. But despite the importance of viscous effects for the study of such systems, a robust and mathematically sound theory of relativistic fluids with viscosity is still lacking. This is due, in part, to difficulties to preserve causality upon the inclusion of viscous and dissipative effects into theories of relativistic fluids. In this talk, we will survey the history of the problem and report on a new approach to relativistic viscous fluids that addresses these issues.

3:30 pm–4:00 pm

Break

4:00 pm–5:00 pm

Maxime van de Moortel, Princeton (hybrid: in person & virtual)

Title: Black holes: the inside story of gravitational collapse

Abstract: What is inside a dynamical black hole? While the local region near time-like infinity is understood for various models, the global structure of the black hole interior has largely remained unexplored. These questions are deeply connected to the nature of singularities in General Relativity and celebrated problems such as Penrose’s Strong Cosmic Censorship Conjecture. I will present my recent resolution of these problems in spherical gravitational collapse, based on the discovery of a novel phenomenon: the breakdown of weak singularities and the dynamical formation of a strong singularity.

Friday, April 8, 2022

Time (ET)

Speaker

Title/Abstract

9:30 am–10:30 am

Ye-Kai Wang, National Cheng Kun University, Taiwan (virtual)

Title: Supertranslation invariance of angular momentum at null infinity in double null gauge

Abstract: This talk accompanies Po-Ning Chen’s talk on Monday with the results described in the double null gauge rather than Bondi-Sachs coordinates. Besides discussing how Chen-Wang-Yau angular momentum resolves the supertranslation ambiguity, we also review the definition of angular momentum defined by A. Rizzi. The talk is based on the joint work with Po-Ning Chen, Jordan Keller, Mu-Tao Wang, and Shing-Tung Yau.

10:30 am–11:30 am

Zoe Wyatt, King’s College London (virtual)

Title: Global Stability of Spacetimes with Supersymmetric Compactifications

Abstract: Spacetimes with compact directions which have special holonomy, such as Calabi-Yau spaces, play an important role in supergravity and string theory. In this talk I will discuss a recent work with Lars Andersson, Pieter Blue and Shing-Tung Yau, where we show the global, nonlinear stability a spacetime which is a cartesian product of a high dimensional Minkowski space with a compact Ricci flat internal space with special holonomy. This stability result is related to a conjecture of Penrose concerning the validity of string theory. Our proof uses the intersection of methods for quasilinear wave and Klein-Gordon equations, and so towards the end of the talk I will also comment more generally on coupled wave–Klein-Gordon equations.

11:30 am–12:30 pm

Elena Giorgi, Columbia University (hybrid: in person & virtual)

Title: The stability of charged black holes

Abstract: Black hole solutions in General Relativity are parametrized by their mass, spin and charge. In this talk, I will motivate why the charge of black holes adds interesting dynamics to solutions of the Einstein equation thanks to the interaction between gravitational and electromagnetic radiation. Such radiations are solutions of a system of coupled wave equations with a symmetric structure which allows to define a combined energy-momentum tensor for the system. Finally, I will show how this physical-space approach is resolutive in the most general case of Kerr-Newman black hole, where the interaction between the radiations prevents the separability in modes.

12:30 pm–1:30 pm

Break

1:30 pm–2:30 pm

Marcus Khuri, Stony Brook University (virtual)

Title: The mass-angular momentum inequality for multiple black holes Abstract: Consider a complete 3-dimensional initial data set for the Einstein equations which has multiple asymptotically flat or asymptotically cylindrical ends. If it is simply connected, axisymmetric, maximal, and satisfies the appropriate energy condition then the ADM mass of any of the asymptotically flat ends is bounded below by the square root of the total angular momentum. This generalizes previous work of Dain, Chrusciel-Li-Weinstein, and Schoen-Zhou which treated either the single black hole case or the multiple black hole case without an explicit lower bound. The proof relies on an analysis of the asymptotics of singular harmonic maps from R^3 \ \Gamma –>H^2 where \Gamma is a coordinate axis. This is joint work with Q. Han, G. Weinstein, and J. Xiong.

2:30 pm–3:30 pm

Martin Lesourd, Harvard (hybrid: in person & virtual)

Title: A Snippet on Mass and the Topology and Geometry of Positive Scalar Curvature

Abstract: I will talk about a small corner of the study of Positive Scalar Curvature (PSC) and questions which are most closely related to the Positive Mass Theorem. The classic questions are ”which topologies allow for PSC?” and ”what is the geometry of manifolds with PSC?”. This is based on joint work with Prof. S-T. Yau, Prof. D. A. Lee, and R. Unger.

3:30 pm–4:00 pm

Break

4:00 pm–5:00 pm

Georgios Moschidis, Princeton (virtual)

Title: Weak turbulence for the Einstein–scalar field system.

Abstract: In the presence of confinement, the Einstein field equations are expected to exhibit turbulent dynamics. In the presence of a negative cosmological constant, the AdS instability conjecture claims the existence of arbitrarily small perturbations to the initial data of Anti-de Sitter spacetime which, under evolution by the vacuum Einstein equations with reflecting boundary conditions at conformal infinity, lead to the formation of black holes after sufficiently long time. In this talk, I will present a rigorous proof of the AdS instability conjecture in the setting of the spherically symmetric Einstein-scalar field system. The construction of the unstable initial data will require carefully designing a family of initial configurations of localized matter beams and estimating the exchange of energy taking place between interacting beams over long periods of time, as well as estimating the decoherence rate of those beams.

For the full schedule, please see: https://cmsa.fas.harvard.edu/gr-program/

This conference will be held virtually on Zoom. Registration is required. Webinar Registration

A few talks will be held in hybrid formats, with talks given from the CMSA seminar room, G-10. Advanced registration for in-person components is required. In-Person Registration

For more information, please see https://cmsa.fas.harvard.edu/gr-program/

The Harvard CMSA will be hosting a conference on General Relativity from April 4-8, 2022.

This conference will be held virtually on Zoom. Registration is required. Webinar Registration A few talks will be held in hybrid formats, with talks given from the CMSA seminar room, G-10. Advanced registration for in-person components is required. In-Person Registration

For more information, please see: https://cmsa.fas.harvard.edu/gr-program/

In this talk I will discuss work in progress in which we classify topological 4-manifolds with boundary and fundamental group Z, under some mild assumptions on the boundary. We apply this classification to provide an algebraic classification of surfaces in simply-connected 4-manifolds with 3-sphere boundary, where the fundamental group on the surface complement is Z. We also compare these homeomorphism classifications with the smooth setting, showing for example that every Hermitian form over the ring of integer Laurent polynomials arises as the equivariant intersection form of a pair of exotic smooth 4-manifolds with boundary and fundamental group Z. This work is joint with Anthony Conway and Mark Powell.

There is a close connection between the scattering amplitudes in planar N=4 SYM theory and the cells in the positive Grassmannian. In the context of BCFW recursion relations the tree-level S-matrix is represented as a sum of planar on-shell diagrams (aka plabic graphs) and associated with logarithmic forms on the Grassmannian cells of certain dimensionality. In this talk, we explore non-adjacent BCFW shifts which naturally lead to non-planar on-shell diagrams and new interesting subspaces inside the real Grassmannian.

The extrinsic geometry of the canonical model of a nonhyperelliptic curve captures many aspects of the intrinsic geometry of the curve. In this talk I will discuss joint work with Izzet Coskun and Eric Larson in which we show that the normal bundle of a general canonical curve of genus at least 7 is always semistable. This makes substantial progress towards a conjecture of Aprodu–Farkas–Ortega, and answers it completely in a third of all cases.

The standard definition of symmetries of a structure given on a set S (in the sense of Bourbaki) is the group of bijective maps S to S, compatible with this structure.

But in fact, symmetries of various structures related to storing and transmitting information (such as information spaces) are naturally embodied in various classes of loops such as Moufang loops, – nonassociative analogs of groups.

The idea of symmetry as a group is closely related to classical physics, in a very definite sense, going back at least to Archimedes. When quantum physics started to replace classical, it turned out that classical symmetries must also be replaced by their quantum versions, e.g. quantum groups.

In this talk we explain how to define and study quantum versions of symmetries, relevant to information theory and other contexts.

About 2 years ago, I have given a new construction of the Euler system of cyclotomic units via Eisenstein congruences in which the p-adic Langlands correspondence for GL_2(\Q_p) plays a central role. In this talk, I want to explain how one can extend this method to obtain a large class of new Euler systems attached to ordinary automorphic forms. This is a work in progress.

It has been known that the four-dimensional abelian chiral gauge theories of an anomaly-free set of Wely fermions can be formulated on the lattice preserving the exact gauge invariance and the required locality property in the framework of the Ginsparg- Wilson relation. This holds true in two dimensions. However, in the related formulation including the mirror Ginsparg-Wilson fermions, it has been argued that the mirror fermions do not decouple: in the 3450 model with Dirac- and Majorana-Yukawa couplings to XY-spin field, the two- point vertex function of the (external) gauge field in the mirror sector shows a singular non-local behavior in the so-called ParaMagnetic Strong-coupling(PMS) phase.

We re-examine why the attempt seems a “Mission: Impossible” in the 3450 model. We point out that the effective operators to break the fermion number symmetries (’t Hooft operators plus others) in the mirror sector do not have sufficiently strong couplings even in the limit of large Majorana-Yukawa couplings. We also observe that the type of Majorana-Yukawa term considered there is singular in the large limit due to the nature of the chiral projection of the Ginsparg-Wilson fermions, but a slight modification without such singularity is allowed by virtue of the very nature.

We then consider a simpler four-flavor axial gauge model, the 14(-1)4 model, in which the U(1)A gauge and Spin(6)( SU(4)) global symmetries prohibit the bilinear terms, but allow the quartic terms to break all the other continuous mirror-fermion symmetries. This model in the weak gauge-coupling limit is related to the eight-flavor Majorana Chain with a reduced SO(6)xSO(2) symmetry in Euclidean path-integral formulation. We formulate the model so that it is well-behaved and simplified in the strong-coupling limit of the quartic operators. Through Monte-Carlo simulations in the weak gauge-coupling limit, we show a numerical evidence that the two-point vertex function of the gauge field in the mirror sector shows a regular local behavior.

Finally, by gauging a U(1) subgroup of the U(1)A× Spin(6)(SU(4)) of the previous model, we formulate the 21(−1)3 chiral gauge model and argue that the induced effective action in the mirror sector satisfies the required locality property. This gives us “A New Hope” for the mission to be accomplished.

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A main challenge in analyzing single-cell RNA sequencing (scRNA-seq) data is to reduce technical variations yet retain cell heterogeneity. Due to low mRNAs content per cell and molecule losses during the experiment (called ‘dropout’), the gene expression matrix has a substantial amount of zero read counts. Existing imputation methods treat either each cell or each gene as independently and identically distributed, which oversimplifies the gene correlation and cell type structure. We propose a statistical model-based approach, called SIMPLEs (SIngle-cell RNA-seq iMPutation and celL clustErings), which iteratively identifies correlated gene modules and cell clusters and imputes dropouts customized for individual gene module and cell type. Simultaneously, it quantifies the uncertainty of imputation and cell clustering via multiple imputations. In simulations, SIMPLEs performed significantly better than prevailing scRNA-seq imputation methods according to various metrics. By applying SIMPLEs to several real datasets, we discovered gene modules that can further classify subtypes of cells. Our imputations successfully recovered the expression trends of marker genes in stem cell differentiation and can discover putative pathways regulating biological processes.

I will explain a mechanism to cancel the vacuum energy and both terms in the Weyl anomaly in the standard model of particle physics, using conformally-coupled dimension-zero scalar fields. Remarkably, given the standard model gauge group SU(3)xSU(2)xU(1), the cancellation requires precisely 48 Weyl spinors — i.e. three generations of standard model fermions, including right-handed neutrinos. Moreover, the scalars possess a scale-invariant power spectrum, suggesting a new explanation for the observed primordial density perturbations in cosmology (without the need for inflation).

As context, I will also introduce a related cosmological picture in which this cancellation mechanism plays an essential role. Our universe seems to be dominated by radiation at early times, and positive vacuum energy at late times. Taking the symmetry and analyticity properties of such a universe seriously suggests a picture in which spacetime has two sheets, related by a symmetry that, in turn, selects a preferred (CPT-symmetric) vacuum state for the quantum fields that live on the spacetime. This line of thought suggests new explanations for a number of observed properties of the universe, including: its homogeneity, isotropy and flatness; the arrow of time; several properties of the primordial perturbations; and the nature of dark matter (which, in this picture, is a right-handed neutrino, radiated from the early universe like Hawking radiation from a black hole). It also makes a number of testable predictions.

(Based on recent, and ongoing, work with Neil Turok: arXiv:1803.08928, arXiv:2109.06204, arXiv:2110.06258, arXiv:2201.07279.)

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Let X be a projective variety and let f be a dominant endomorphism of f, both of which are defined over a number field K. We consider the question of when there is some integer n, depending only on X and K, such that whenever x and y are K-points of X with the property that some iterate of f maps x and y to the same point, we necessarily have that the n-th iterate of f also achieves this. We consider this an instance of “dynamical cancellation’’ and we show that such a cancellation result holds for etale morphisms of projective varieties as well as self-maps of smooth projective curves. As a result we are able to prove a general cancellation result for semigroups of polynomials: if f_1, … , f_r are polynomials of degree at least two then there is a proper closed subset of P^1 x P^1 with the property that for any a, b in K satisfying phi(a)=phi(b) for some phi in the semigroup generated by f_1, … f_r under composition, we necessarily have (a,b) lies in this closed subset. Moreover, we show that this Z can be taken to be the union of the diagonal with a finite set of points for “non-exceptional” semigroups. This is joint work with Matt Satriano and Yohsuke Matsusawa.

On April 15, 2022, the CMSA will hold a one-day workshop, Machine Learning and Mathematical Conjecture, related to the New Technologies in Mathematics Seminar Series. Organizers: Michael R. Douglas (CMSA/Stony Brook/IAIFI) and Peter Chin (CMSA/BU).

Machine learning has driven many exciting recent scientific advances. It has enabled progress on long-standing challenges such as protein folding, and it has helped mathematicians and mathematical physicists create new conjectures and theorems in knot theory, algebraic geometry, and representation theory.

At this workshop, we will bring together mathematicians, theoretical physicists and machine learning researchers to review the state of the art in machine learning, discuss how ML results can be used to inspire, test and refine precise conjectures, and identify mathematical questions which may be suitable for this approach.

Speakers:

James Halverson, Northeastern University Dept. of Physics and IAIFI

Fabian Ruehle, Northeastern University Dept. of Physics and Mathematics and IAIFI

Andrew Sutherland, MIT Department of Mathematics

9:30 am – 10:20 am

James Halverson: Machine Learning for Mathematicians

10:30 am – 11:20 am

Andrew Sutherland: Number Theory

11:30 am – 12:20 pm

Fabian Ruehle: Knot Theory

Lunch break

2:00 pm –3:30 pm

Computer demonstrations

3:45 pm – 4:45 pm

Discussion

The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA.

On April 18, 20, and 22, Eric Maskin (Harvard University) will give three introductory lectures on Game Theory.

April 18, 2022 | 9:30–11:00 am ET Title: Game Theory Basics and Classical Existence Theorems Abstract: Games in extensive and normal form. Equilibrium existence theorems by Nash, von Neumann, and Zermelo

April 20, 2022 | 9:30–11:00 am ET Title: Mechanism Design Abstract: Given a social goal, under what circumstances can we design a game to achieve that goal?

April 22, 2022 | 9:30–11:00 am ET Title: Auction Theory Abstract: Equivalences among four standard auctions: the high-bid auction (the high bidder wins and pays her bid); the second-bid auction (the high bidder wins and pays the second-highest bid); the Dutch auction (the auctioneer lowers the price successively until some bidder is willing to pay); and the English auction (bidders raise their bids successively until no one wants to bid higher).

In studying complex Chern-Simons theory on a Seifert manifold, Gukov-Pei proposed an equivariant Verlinde formula, a one-parameter deformation of the celebrated Verlinde formula. It computes, among many things, the graded dimension of the space of holomorphic sections of (powers of) a natural determinant line bundle over the Hitchin moduli space. Gukov-Pei conjectured that the equivariant Verlinde numbers are equal to the equivariant quantum K-invariants of a non-compact (Kahler) quotient space studied by Hanany-Tong.

In this talk, I will explain the setup of this conjecture and its proof via wall-crossing of moduli spaces of (parabolic) Bradlow-Higgs triples. It is based on work in progress with Wei Gu and Du Pei.

For information on how to join, please see: https://cmsa.fas.harvard.edu/seminars-and-colloquium/

We will explain an isoperimetric-type inequality for subsets of the Heisenberg group that controls the size of the (suitably defined) vertical boundary by the perimeter. This inequality was conceived for a specific application that we will describe, but in the course of its investigation there were twists and turns that unearthed unexpected geometric phenomena and led to further applications to longstanding questions. These will be explained, but the main purpose of this talk is to show the geometric structures that arise in the proof.

Wilson loop diagrams can be used to study amplitudes in N=4 SYM. I will set them up and talk about some of their combinatorial aspects, such as how many Wilson loop diagrams give the same positroid and how to combinatorially read off the dimension and the denominators for the integrands.

This seminar will be held in person and online via Zoom.

To register for the in person seminar, please see: https://forms.gle/rMHV1fQP6Lu576eH9

For information on how to join, please see: https://cmsa.fas.harvard.edu/seminars-and-colloquium/

On April 18, 20, and 22, Eric Maskin (Harvard University) will give three introductory lectures on Game Theory.

April 18, 2022 | 9:30–11:00 am ET Title: Game Theory Basics and Classical Existence Theorems Abstract: Games in extensive and normal form. Equilibrium existence theorems by Nash, von Neumann, and Zermelo

April 20, 2022 | 9:30–11:00 am ET Title: Mechanism Design Abstract: Given a social goal, under what circumstances can we design a game to achieve that goal?

April 22, 2022 | 9:30–11:00 am ET Title: Auction Theory Abstract: Equivalences among four standard auctions: the high-bid auction (the high bidder wins and pays her bid); the second-bid auction (the high bidder wins and pays the second-highest bid); the Dutch auction (the auctioneer lowers the price successively until some bidder is willing to pay); and the English auction (bidders raise their bids successively until no one wants to bid higher).

Since its discovery, unconventional superconductivity in cuprates has motivated the search for materials with analogous electronic or atomic structure. We have used soft chemistry approaches to synthesize superconducting infinite layer nickelates from their perovskite precursor phase. We will present the synthesis and transport properties of the nickelates, observation of a doping-dependent superconducting dome, and our current understanding of their electronic and magnetic structure.

We prove that with high probability $G^{(3)}(n,n^{-1+o(1)})$ contains a spanning Steiner triple system. We also prove the analogous result for spanning Latin squares. This threshold is sharp up to a subpolynomial factor. Our result follows from a novel bootstrapping scheme that utilizes iterative absorption as well as recent connections which have been established between thresholds and spread measures. Joint work with Ashwin Sah and Michael Simkin.

It has been proved that there is no general algorithm for finding all the integer solutions to a multivariable polynomial equation. Nevertheless, there are some methods that succeed if the equation has a certain form. I will explain one such method. If you want to try to rediscover it yourself, try to provably find all the integer solutions to y^2 = x^4 + 4x^3 – 2x^2 + 6x + 20.

*This week the seminar will be held at the Science Center North Lawn

Encryption is the backbone of cybersecurity. While encryption can secure data both in transit and at rest, in the new era of ubiquitous computing, modern cryptography also aims to protect data during computation. Secure multi-party computation (MPC) is a powerful technology to tackle this problem, which enables distrustful parties to jointly perform computation over their private data without revealing their data to each other. Although it is theoretically feasible and provably secure, the adoption of MPC in real industry is still very much limited as of today, the biggest obstacle of which boils down to its efficiency.

My research goal is to bridge the gap between the theoretical feasibility and practical efficiency of MPC. Towards this goal, my research spans both theoretical and applied cryptography. In theory, I develop new techniques for achieving general MPC with the optimal complexity, bringing theory closer to practice. In practice, I design tailored MPC to achieve the best concrete efficiency for specific real-world applications. In this talk, I will discuss the challenges in both directions and how to overcome these challenges using cryptographic approaches. I will also show strong connections between theory and practice.

For information on how to join, please see: https://cmsa.fas.harvard.edu/seminars-and-colloquium/

Abstract: As machine learning plays a more prominent role in our society, we need learning algorithms that are reliable and robust. It is important to understand whether existing algorithms are robust against adversarial attacks and design new robust solutions that work under weaker assumptions. The long-term goal is to bridge the gap between the growing need for robust algorithms and the lack of systematic understanding of robustness.

In this talk, I will discuss the challenges that arise in the design and analysis of robust algorithms for machine learning. I will focus on three lines of my recent work: (1) designing faster and simpler algorithms for high-dimensional robust statistics where a small fraction of the input data is arbitrarily corrupted, (2) analyzing the optimization landscape of non-convex approaches for low-rank matrix problems and making non-convex optimization robust against semi-random adversaries, and (3) considering learning in the presence of strategic behavior where the goal is to design good algorithms that account for the agents’ strategic responses.

Bio: Yu Cheng is an assistant professor in the Mathematics department at the University of Illinois at Chicago. He obtained his Ph.D. in Computer Science from the University of Southern California. Before joining UIC, he was a postdoc at Duke University and a visiting member at the Institute for Advanced Study. His main research interests include machine learning, optimization, and game theory.

For information on how to join, please see: https://cmsa.fas.harvard.edu/seminars-and-colloquium/

On April 18, 20, and 22, Eric Maskin (Harvard University) will give three introductory lectures on Game Theory.

April 18, 2022 | 9:30–11:00 am ET Title: Game Theory Basics and Classical Existence Theorems Abstract: Games in extensive and normal form. Equilibrium existence theorems by Nash, von Neumann, and Zermelo

April 20, 2022 | 9:30–11:00 am ET Title: Mechanism Design Abstract: Given a social goal, under what circumstances can we design a game to achieve that goal?

April 22, 2022 | 9:30–11:00 am ET Title: Auction Theory Abstract: Equivalences among four standard auctions: the high-bid auction (the high bidder wins and pays her bid); the second-bid auction (the high bidder wins and pays the second-highest bid); the Dutch auction (the auctioneer lowers the price successively until some bidder is willing to pay); and the English auction (bidders raise their bids successively until no one wants to bid higher).

The Eulerian idempotents of the symmetric group generate a family of representations—the Eulerian representations—that have connections to configuration spaces, equivariant cohomology, and Solomon’s descent algebra. These representations are defined in terms of S_n, but can be “lifted” to representations of S_{n+1} called the Whitehouse representations. I will describe this story in detail and present recent work generalizing it to the hyperoctahedral group (e.g. Type B). In this setting, configuration spaces will be replaced by certain orbit configuration spaces and Solomon’s descent algebra is replaced by the Mantaci-Reutenauer algebra. All of the above will be defined in the talk, which is based on the preprint https://arxiv.org/abs/2203.09504.

The Higgs phase of a gauge theory is important to both fundamental physics (e.g., electroweak theory) as well as condensed matter systems (superconductors and other emergent phenomena). However, such a charge condensate seems subtle and is sometimes described as the spontaneous breaking of gauge symmetry (or a global subgroup). In this talk, I will argue that the Higgs phase is best understood as a symmetry-protected topological (SPT) phase. The concept of SPT phases arose out of the condensed matter community, to describe systems with short-range entanglement and edge modes which cannot be removed in the presence of certain symmetries. The perspective that the Higgs phase is an SPT phase recovers known properties of the Higgs phase and provides new insights. In particular, we revisit the Fradkin-Shenker model and the distinction between the Higgs and confined phases of a gauge theory.

For information on how to join, please see: https://cmsa.fas.harvard.edu/seminars-and-colloquium/

We discuss the algebraic geometry of maximum likelihood estimation from the perspective of scattering amplitudes in particle physics. A guiding examples the moduli space of n-pointed rational curves. The scattering potential plays the role of the log-likelihood function, and its critical points are solutions to rational function equations. Their number is an Euler characteristic. Soft limit degenerations are combined with certified numerical methods for concrete computations.

This seminar will be held in person and online via Zoom.

To register for the in person seminar, please see: https://forms.gle/rMHV1fQP6Lu576eH9

For information on how to join, please see: https://cmsa.fas.harvard.edu/seminars-and-colloquium/

In ‘Mirror symmetry for log Calabi–Yau surfaces I’, given a smooth log Calabi–Yau surface pair (Y,D), Gross–Hacking–Keel constructed its mirror family as the spectrum of an explicit algebra whose structure coefficients are determined by the enumerative geometry of (Y,D). As a follow-up of the work of Gross–Hacking–Keel, when (Y,D) is positive, we prove the modularity of the mirror family as the universal family of log Calabi-Yau surface pairs deformation equivalent to (Y,D) with at worst du Val singularities. As a corollary, we show that the ring of regular functions of a smooth affine log Calabi–Yau surface has a canonical basis of theta functions. The key step towards the proof of the main theorem is the application of the tropical construction of singular cycles and explicit formulas of period integrals given in the work of Helge–Siebert. This is joint work with Jonathan Lai.

For information on how to join, please see: https://cmsa.fas.harvard.edu/seminars-and-colloquium/

For a prime number p, the crystalline cohomology of an F_p-scheme can be regarded as an analogue of the singular cohomology with Z_p coefficients of a topological space. On the topological side, there are other “generalized” cohomology theories, e.g. K-theory and cobordism, and these are related to natural operations on singular cohomology. In this talk, I will discuss analogues of these generalized cohomology theories and cohomology operations in the crystalline setting.

On April 27–29, 2022, the CMSA will host a workshop on Nonlinear Algebra and Combinatorics.

Organizers: Bernd Sturmfels (MPI Leipzig) and Lauren Williams (Harvard).

In recent years, ideas from integrable systems and scattering amplitudes have led to advances in nonlinear algebra and combinatorics. In this short workshop, aimed at younger participants in the field, we will explore some of the interactions between the above topics.

The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA. For a list of lodging options convenient to the Center, please visit our recommended lodgings page.

All non-Harvard affiliated visitors to the CMSA building are required to complete this covid form prior to arrival: https://forms.gle/xKykcNcXq7ciZuvJ8

Abstract: Matroid theory provides a combinatorial model for linearity, but it plays useful roles beyond linearity. In the classical setup, a linear subspace V of an n-dimensional vector space gives rise to a matroid M(V) on {1,…,n}. However, the matroid M(V) also knows about some nonlinear geometric spaces related to V. Conversely, those nonlinear spaces teach us things we didn’t know about matroids. My talk will discuss some examples.

10:30 am–11:00 am

COFFEE BREAK

11:00 am–11:45 am

Chris Eur

Title: Tautological classes of matroids

Abstract: Algebraic geometry has furnished fruitful tools for studying matroids, which are combinatorial abstractions of hyperplane arrangements. We first survey some recent developments, pointing out how these developments remained partially disjoint. We then introduce certain vector bundles (K-classes) on permutohedral varieties, which we call “tautological bundles (classes)” of matroids, as a new framework that unifies, recovers, and extends these recent developments. Our framework leads to new questions that further probe the boundary between combinatorics and geometry. Joint work with Andrew Berget, Hunter Spink, and Dennis Tseng.

11:45 am–2:00 pm

LUNCH BREAK

2:00 pm–2:45 pm

Nick Early

Title: Biadjoint Scalars and Associahedra from Residues of Generalized Amplitudes

Abstract: The associahedron is known to encapsulate physical properties such as the notion of tree-level factorization for one of the simplest Quantum Field Theories, the biadjoint scalar, which has only cubic interactions. I will discuss novel instances of the associahedron and the biadjoint scalar in a class of generalized amplitudes, discovered by Cachazo, Early, Guevara and Mizera, by taking certain limits thereof. This connection leads to a simple proof of a new realization of the associahedron involving a Minkowski sum of certain positroid polytopes in the second hypersimplex.

2:45 pm–3:30 pm

Anna Seigal

Title: Invariant theory for maximum likelihood estimation

Abstract: I will talk about work to uncover connections between invariant theory and maximum likelihood estimation. I will describe how norm minimization over a torus orbit is equivalent to maximum likelihood estimation in log-linear models. We will see the role played by polytopes and discuss connections to scaling algorithms. Based on joint work with Carlos Améndola, Kathlén Kohn, and Philipp Reichenbach.

3:30 pm–4:00 pm

COFFEE BREAK

4:00 pm–4:45 pm

Matteo Parisi

Title: Amplituhedra, Scattering Amplitudes, and Triangulations

Abstract: In this talk I will discuss about Amplituhedra – generalizations of polytopes inside the Grassmannian – introduced by physicists to encode interactions of elementary particles in certain Quantum Field Theories. In particular, I will explain how the problem of finding triangulations of Amplituhedra is connected to computing scattering amplitudes of N=4 super Yang-Mills theory. Triangulations of polygons are encoded in the associahedron, studied by Stasheff in the sixties; in the case of polytopes, triangulations are captured by secondary polytopes, constructed by Gelfand et al. in the nineties. Whereas a “secondary” geometry describing triangulations of Amplituhedra is still not known, and we pave the way for such studies. I will discuss how the combinatorics of triangulations interplays with T-duality from String Theory, in connection with the Momentum Amplituhedron. A generalization of T-duality led us to discover a striking duality between Amplituhedra of “m=2” type and a seemingly unrelated object – the Hypersimplex. The latter is a polytope which appears in many contexts, from matroid theory to tropical geometry. Based on joint works with Lauren Williams, Melissa Sherman-Bennett, Tomasz Lukowski.

4:45 pm–5:30 pm

Melissa Sherman-Bennett

Title: The hypersimplex and the m=2 amplituhedron

Abstract: In this talk, I’ll continue where Matteo left off. I’ll give some more details about the curious correspondence between the m=2 amplituhedron, a 2k-dimensional subset of Gr(k, k+2), and the hypersimplex, an (n-1)-dimensional polytope in R^n. The amplituhedron and hypersimplex are both images of the totally nonnegative Grassmannian under some map (the amplituhedron map and the moment map, respectively), but are different dimensions and live in very different ambient spaces. I’ll talk about joint work with Matteo Parisi and Lauren Williams in which we give a bijection between decompositions of the amplituhedron and decompositions of the hypersimplex (originally conjectured by Lukowski–Parisi–Williams). The hypersimplex decompositions are closely related to matroidal subdivisions. Along the way, we prove a nice description of the m=2 amplituhedron conjectured by Arkani-Hamed–Thomas–Trnka and give a new decomposition of the m=2 amplituhedron into Eulerian-number-many chambers, inspired by an analogous triangulation of the hypersimplex into Eulerian-number-many simplices.

Thursday, April 28, 2022

9:30 am–10:30 am

Claudia Fevola

Title: Nonlinear Algebra meets Feynman integrals

Abstract: Feynman integrals play a central role in particle physics in the theory of scattering amplitudes. They form a finite-dimensional vector space and the elements of a basis are named “master integrals” in the physics literature. The number of master integrals has been interpreted in different ways: it equals the dimension of a twisted de Rham cohomology group, the Euler characteristic of a very affine variety, and the holonomic rank of a D-module. In this talk, we are interested in a more general family of integrals that contains Feynman integrals as a special case. We explore this setting using tools coming from nonlinear algebra. This is an ongoing project with Daniele Agostini, Anna-Laura Sattelberger, and Simon Telen.

10:30 am–11:00 am

COFFEE BREAK

11:00 am–11:45 am

Simon Telen

Title: Landau discriminants

Abstract: The Landau discriminant is a projective variety containing kinematic parameters for which a Feynman integral can have singularities. We present a definition and geometric properties. We discuss how to compute Landau discriminants using symbolic and numerical methods. Our methods can be used, for instance, to compute the Landau discriminant of the pentabox diagram, which is a degree 12 hypersurface in 6-space. This is joint work with Sebastian Mizera.

11:45 am–2:00 pm

LUNCH BREAK

2:00 pm–2:45 pm

Christian Gaetz

Title: 1-skeleton posets of Bruhat interval polytopes

Abstract: Bruhat interval polytopes are a well-studied class of generalized permutohedra which arise as moment map images of various toric varieties and totally positive spaces in the flag variety. I will describe work in progress in which I study the 1-skeleta of these polytopes, viewed as posets interpolating between weak order and Bruhat order. In many cases these posets are lattices and the polytopes, despite not being simple, have interesting h-vectors. In a special case, work of Williams shows that Bruhat interval polytopes are isomorphic to bridge polytopes, so that chains in the 1-skeleton poset correspond to BCFW-bridge decompositions of plabic graphs.

2:45 pm–3:30 pm

Madeleine Brandt

Title: Top Weight Cohomology of $A_g$

Abstract: I will discuss a recent project in computing the top weight cohomology of the moduli space $A_g$ of principally polarized abelian varieties of dimension $g$ for small values of $g$. This piece of the cohomology is controlled by the combinatorics of the boundary strata of a compactification of $A_g$. Thus, it can be computed combinatorially. This is joint work with Juliette Bruce, Melody Chan, Margarida Melo, Gwyneth Moreland, and Corey Wolfe.

3:30 pm–4:00 pm

COFFEE BREAK

4:00 pm–5:00 pm

Emma Previato

Title: Sigma function on curves with non-symmetric semigroup

Abstract: We start with an overview of the correspondence between spectral curves and commutative rings of differential operators, integrable hierarchies of non-linear PDEs and Jacobian vector fields. The coefficients of the operators can be written explicitly in terms of the Kleinian sigma function: Weierstrass’ sigma function was generalized to genus greater than one by Klein, and is a ubiquitous tool in integrability. The most accessible case is the sigma function of telescopic curves. In joint work with J. Komeda and S. Matsutani, we construct a curve with non-symmetric Weierstrass semigroup (equivalently, Young tableau), consequently non-telescopic, and its sigma function. We conclude with possible applications to commutative rings of differential operators.

Abstract: It is well-known that soliton solutions of the KdV hierarchy are obtained by singular limits of hyper-elliptic curves. However, there is no general results for soliton solutions of the KP hierarchy, KP solitons. In this talk, I will show that some of the KP solitons are related to the singular space curves associated with certain class of numerical semigroups.

10:00 am–10:30 am

COFFEE BREAK

10:30 am–11:15 am

Yelena Mandelshtam

Title: Curves, degenerations, and Hirota varieties

Abstract: The Kadomtsev-Petviashvili (KP) equation is a differential equation whose study yields interesting connections between integrable systems and algebraic geometry. In this talk I will discuss solutions to the KP equation whose underlying algebraic curves undergo tropical degenerations. In these cases, Riemann’s theta function becomes a finite exponential sum that is supported on a Delaunay polytope. I will introduce the Hirota variety which parametrizes all KP solutions arising from such a sum. I will then discuss a special case, studying the Hirota variety of a rational nodal curve. Of particular interest is an irreducible subvariety that is the image of a parameterization map. Proving that this is a component of the Hirota variety entails solving a weak Schottky problem for rational nodal curves. This talk is based on joint work with Daniele Agostini, Claudia Fevola, and Bernd Sturmfels.

11:15 am–12:00 pm

Charles Wang

Title: Differential Algebra of Commuting Operators

Abstract: In this talk, we will give an overview of the problem of finding the centralizer of a fixed differential operator in a ring of differential operators, along with connections to integrable hierarchies and soliton solutions to e.g. the KdV or KP equations. Given these interesting connections, it is important to be able to compute centralizers of differential operators, and we discuss how to use techniques from differential algebra to approach this question, as well as how having these computational tools can help in understanding the structure of soliton solutions to these equations.

12:00 pm–2:00 pm

LUNCH BREAK

2:00 pm–3:00 pm

Sebastian Mizera

Title: Feynman Polytopes

Abstract: I will give an introduction to a class of polytopes that recently emerged in the study of scattering amplitudes in quantum field theory.

3:00 pm–3:30 pm

COFFEE BREAK

3:30 pm–4:30 pm

Nima Arkani-Hamed

Title: Spacetime, Quantum Mechanics and Combinatorial Geometries at Infinity

In recent years, ideas from integrable systems and scattering amplitudes have led to advances in nonlinear algebra and combinatorics. In this short workshop, aimed at younger participants in the field, we will explore some of the interactions between the above topics.

Invited speakers include Federico Ardila, Nima Arkani-Hamed, Madeleine Brandt, Freddy Cachazo, Chris Eur, Claudia Fevola, Christian Gaetz, Yuji Kodama, Fatemeh Mohammadi, Matteo Parisi, Anna Seigal, Melissa Sherman-Bennett, Simon Telen, and Charles Wang.

The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA

Suppose we transmit n bits on a noisy channel that deletes some fraction of the bits arbitrarily. What’s the supremum p* of deletion fractions that can be corrected with a binary code of non-vanishing rate? Evidently p* is at most 1/2 as the adversary can delete all occurrences of the minority bit. It was unknown whether this simple upper bound could be improved, or one could in fact correct deletion fractions approaching 1/2.

We show that there exist absolute constants A and delta > 0 such that any subset of n-bit strings of size exp((log n)^A) must contain two strings with a common subsequence of length (1/2+delta)n. This immediately implies that the zero-rate threshold p* of worst-case bit deletions is bounded away from 1/2.

Our techniques include string regularity arguments and a structural lemma that classifies bit-strings by their oscillation patterns. Leveraging these tools, we find in any large code two strings with similar oscillation patterns, which is exploited to find a long common subsequence.

This is joint work with Xiaoyu He and Ray Li.

For information on how to join, please see: https://cmsa.fas.harvard.edu/seminars-and-colloquium/

The ghost conjecture predicts slopes of modular forms whose residual representation is locally reducible. In this talk, we’ll examine locally irreducible representations and discuss recent progress on formulating a conjecture in this case. It’s a lot trickier and the story remains incomplete, but we will discuss how an irregular ghost conjecture is intimately related to reductions of crystalline representations.

On April 27–29, 2022, the CMSA will host a workshop on Nonlinear Algebra and Combinatorics.

Organizers: Bernd Sturmfels (MPI Leipzig) and Lauren Williams (Harvard).

In recent years, ideas from integrable systems and scattering amplitudes have led to advances in nonlinear algebra and combinatorics. In this short workshop, aimed at younger participants in the field, we will explore some of the interactions between the above topics.

The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA. For a list of lodging options convenient to the Center, please visit our recommended lodgings page.

All non-Harvard affiliated visitors to the CMSA building are required to complete this covid form prior to arrival: https://forms.gle/xKykcNcXq7ciZuvJ8

Abstract: Matroid theory provides a combinatorial model for linearity, but it plays useful roles beyond linearity. In the classical setup, a linear subspace V of an n-dimensional vector space gives rise to a matroid M(V) on {1,…,n}. However, the matroid M(V) also knows about some nonlinear geometric spaces related to V. Conversely, those nonlinear spaces teach us things we didn’t know about matroids. My talk will discuss some examples.

10:30 am–11:00 am

COFFEE BREAK

11:00 am–11:45 am

Chris Eur

Title: Tautological classes of matroids

Abstract: Algebraic geometry has furnished fruitful tools for studying matroids, which are combinatorial abstractions of hyperplane arrangements. We first survey some recent developments, pointing out how these developments remained partially disjoint. We then introduce certain vector bundles (K-classes) on permutohedral varieties, which we call “tautological bundles (classes)” of matroids, as a new framework that unifies, recovers, and extends these recent developments. Our framework leads to new questions that further probe the boundary between combinatorics and geometry. Joint work with Andrew Berget, Hunter Spink, and Dennis Tseng.

11:45 am–2:00 pm

LUNCH BREAK

2:00 pm–2:45 pm

Nick Early

Title: Biadjoint Scalars and Associahedra from Residues of Generalized Amplitudes

Abstract: The associahedron is known to encapsulate physical properties such as the notion of tree-level factorization for one of the simplest Quantum Field Theories, the biadjoint scalar, which has only cubic interactions. I will discuss novel instances of the associahedron and the biadjoint scalar in a class of generalized amplitudes, discovered by Cachazo, Early, Guevara and Mizera, by taking certain limits thereof. This connection leads to a simple proof of a new realization of the associahedron involving a Minkowski sum of certain positroid polytopes in the second hypersimplex.

2:45 pm–3:30 pm

Anna Seigal

Title: Invariant theory for maximum likelihood estimation

Abstract: I will talk about work to uncover connections between invariant theory and maximum likelihood estimation. I will describe how norm minimization over a torus orbit is equivalent to maximum likelihood estimation in log-linear models. We will see the role played by polytopes and discuss connections to scaling algorithms. Based on joint work with Carlos Améndola, Kathlén Kohn, and Philipp Reichenbach.

3:30 pm–4:00 pm

COFFEE BREAK

4:00 pm–4:45 pm

Matteo Parisi

Title: Amplituhedra, Scattering Amplitudes, and Triangulations

Abstract: In this talk I will discuss about Amplituhedra – generalizations of polytopes inside the Grassmannian – introduced by physicists to encode interactions of elementary particles in certain Quantum Field Theories. In particular, I will explain how the problem of finding triangulations of Amplituhedra is connected to computing scattering amplitudes of N=4 super Yang-Mills theory. Triangulations of polygons are encoded in the associahedron, studied by Stasheff in the sixties; in the case of polytopes, triangulations are captured by secondary polytopes, constructed by Gelfand et al. in the nineties. Whereas a “secondary” geometry describing triangulations of Amplituhedra is still not known, and we pave the way for such studies. I will discuss how the combinatorics of triangulations interplays with T-duality from String Theory, in connection with the Momentum Amplituhedron. A generalization of T-duality led us to discover a striking duality between Amplituhedra of “m=2” type and a seemingly unrelated object – the Hypersimplex. The latter is a polytope which appears in many contexts, from matroid theory to tropical geometry. Based on joint works with Lauren Williams, Melissa Sherman-Bennett, Tomasz Lukowski.

4:45 pm–5:30 pm

Melissa Sherman-Bennett

Title: The hypersimplex and the m=2 amplituhedron

Abstract: In this talk, I’ll continue where Matteo left off. I’ll give some more details about the curious correspondence between the m=2 amplituhedron, a 2k-dimensional subset of Gr(k, k+2), and the hypersimplex, an (n-1)-dimensional polytope in R^n. The amplituhedron and hypersimplex are both images of the totally nonnegative Grassmannian under some map (the amplituhedron map and the moment map, respectively), but are different dimensions and live in very different ambient spaces. I’ll talk about joint work with Matteo Parisi and Lauren Williams in which we give a bijection between decompositions of the amplituhedron and decompositions of the hypersimplex (originally conjectured by Lukowski–Parisi–Williams). The hypersimplex decompositions are closely related to matroidal subdivisions. Along the way, we prove a nice description of the m=2 amplituhedron conjectured by Arkani-Hamed–Thomas–Trnka and give a new decomposition of the m=2 amplituhedron into Eulerian-number-many chambers, inspired by an analogous triangulation of the hypersimplex into Eulerian-number-many simplices.

Thursday, April 28, 2022

9:30 am–10:30 am

Claudia Fevola

Title: Nonlinear Algebra meets Feynman integrals

Abstract: Feynman integrals play a central role in particle physics in the theory of scattering amplitudes. They form a finite-dimensional vector space and the elements of a basis are named “master integrals” in the physics literature. The number of master integrals has been interpreted in different ways: it equals the dimension of a twisted de Rham cohomology group, the Euler characteristic of a very affine variety, and the holonomic rank of a D-module. In this talk, we are interested in a more general family of integrals that contains Feynman integrals as a special case. We explore this setting using tools coming from nonlinear algebra. This is an ongoing project with Daniele Agostini, Anna-Laura Sattelberger, and Simon Telen.

10:30 am–11:00 am

COFFEE BREAK

11:00 am–11:45 am

Simon Telen

Title: Landau discriminants

Abstract: The Landau discriminant is a projective variety containing kinematic parameters for which a Feynman integral can have singularities. We present a definition and geometric properties. We discuss how to compute Landau discriminants using symbolic and numerical methods. Our methods can be used, for instance, to compute the Landau discriminant of the pentabox diagram, which is a degree 12 hypersurface in 6-space. This is joint work with Sebastian Mizera.

11:45 am–2:00 pm

LUNCH BREAK

2:00 pm–2:45 pm

Christian Gaetz

Title: 1-skeleton posets of Bruhat interval polytopes

Abstract: Bruhat interval polytopes are a well-studied class of generalized permutohedra which arise as moment map images of various toric varieties and totally positive spaces in the flag variety. I will describe work in progress in which I study the 1-skeleta of these polytopes, viewed as posets interpolating between weak order and Bruhat order. In many cases these posets are lattices and the polytopes, despite not being simple, have interesting h-vectors. In a special case, work of Williams shows that Bruhat interval polytopes are isomorphic to bridge polytopes, so that chains in the 1-skeleton poset correspond to BCFW-bridge decompositions of plabic graphs.

2:45 pm–3:30 pm

Madeleine Brandt

Title: Top Weight Cohomology of $A_g$

Abstract: I will discuss a recent project in computing the top weight cohomology of the moduli space $A_g$ of principally polarized abelian varieties of dimension $g$ for small values of $g$. This piece of the cohomology is controlled by the combinatorics of the boundary strata of a compactification of $A_g$. Thus, it can be computed combinatorially. This is joint work with Juliette Bruce, Melody Chan, Margarida Melo, Gwyneth Moreland, and Corey Wolfe.

3:30 pm–4:00 pm

COFFEE BREAK

4:00 pm–5:00 pm

Emma Previato

Title: Sigma function on curves with non-symmetric semigroup

Abstract: We start with an overview of the correspondence between spectral curves and commutative rings of differential operators, integrable hierarchies of non-linear PDEs and Jacobian vector fields. The coefficients of the operators can be written explicitly in terms of the Kleinian sigma function: Weierstrass’ sigma function was generalized to genus greater than one by Klein, and is a ubiquitous tool in integrability. The most accessible case is the sigma function of telescopic curves. In joint work with J. Komeda and S. Matsutani, we construct a curve with non-symmetric Weierstrass semigroup (equivalently, Young tableau), consequently non-telescopic, and its sigma function. We conclude with possible applications to commutative rings of differential operators.

Abstract: It is well-known that soliton solutions of the KdV hierarchy are obtained by singular limits of hyper-elliptic curves. However, there is no general results for soliton solutions of the KP hierarchy, KP solitons. In this talk, I will show that some of the KP solitons are related to the singular space curves associated with certain class of numerical semigroups.

10:00 am–10:30 am

COFFEE BREAK

10:30 am–11:15 am

Yelena Mandelshtam

Title: Curves, degenerations, and Hirota varieties

Abstract: The Kadomtsev-Petviashvili (KP) equation is a differential equation whose study yields interesting connections between integrable systems and algebraic geometry. In this talk I will discuss solutions to the KP equation whose underlying algebraic curves undergo tropical degenerations. In these cases, Riemann’s theta function becomes a finite exponential sum that is supported on a Delaunay polytope. I will introduce the Hirota variety which parametrizes all KP solutions arising from such a sum. I will then discuss a special case, studying the Hirota variety of a rational nodal curve. Of particular interest is an irreducible subvariety that is the image of a parameterization map. Proving that this is a component of the Hirota variety entails solving a weak Schottky problem for rational nodal curves. This talk is based on joint work with Daniele Agostini, Claudia Fevola, and Bernd Sturmfels.

11:15 am–12:00 pm

Charles Wang

Title: Differential Algebra of Commuting Operators

Abstract: In this talk, we will give an overview of the problem of finding the centralizer of a fixed differential operator in a ring of differential operators, along with connections to integrable hierarchies and soliton solutions to e.g. the KdV or KP equations. Given these interesting connections, it is important to be able to compute centralizers of differential operators, and we discuss how to use techniques from differential algebra to approach this question, as well as how having these computational tools can help in understanding the structure of soliton solutions to these equations.

12:00 pm–2:00 pm

LUNCH BREAK

2:00 pm–3:00 pm

Sebastian Mizera

Title: Feynman Polytopes

Abstract: I will give an introduction to a class of polytopes that recently emerged in the study of scattering amplitudes in quantum field theory.

3:00 pm–3:30 pm

COFFEE BREAK

3:30 pm–4:30 pm

Nima Arkani-Hamed

Title: Spacetime, Quantum Mechanics and Combinatorial Geometries at Infinity

Let X be a compact hyperbolic surface. We can see that there is a constant C(X) such that the intersection number of the closed geodesics is \leq C(X) times the product of their lengths. Consider the optimum constant C(X). In this talk, we describe its asymptotic behavior in terms of systole, length of the shortest closed geodesic on X.

In recent years, ideas from integrable systems and scattering amplitudes have led to advances in nonlinear algebra and combinatorics. In this short workshop, aimed at younger participants in the field, we will explore some of the interactions between the above topics.

Invited speakers include Federico Ardila, Nima Arkani-Hamed, Madeleine Brandt, Freddy Cachazo, Chris Eur, Claudia Fevola, Christian Gaetz, Yuji Kodama, Fatemeh Mohammadi, Matteo Parisi, Anna Seigal, Melissa Sherman-Bennett, Simon Telen, and Charles Wang.

The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA

We will overview the program of geometrically engineering four dimensional supersymmetric QFTs as compactifications of six dimensional SCFTs. In particular we will discuss how strong coupling phenomena in four dimensions, such as duality and emergence of symmetry, can be better understood in such geometric constructions.

For information on how to join, please see: https://cmsa.fas.harvard.edu/seminars-and-colloquium/

Active matter describes out-of-equilibrium materials composed of motile building blocks that convert free energy into mechanical work. The continuous input of energy at the particle scale liberates these systems from the constraints of thermodynamic equilibrium, leading to emergent collective behaviors not found in passive materials. In this talk, I will describe our recent efforts to build simple active systems composed of purified proteins and identify generic emergent behaviors in active systems. I will first discuss two distinct activity-driven instabilities in suspensions of microtubules and molecular motors. Second, I will describe a new model system for polar fluid whose collective dynamics are driven by the non-equilibrium turnover of actin filaments. Our results illustrate how biomimetic materials can serve as a platform for studying non-equilibrium statistical mechanics, as well as shine light on the physical mechanisms that regulate self-organization in living matter.

On April 27–29, 2022, the CMSA will host a workshop on Nonlinear Algebra and Combinatorics.

Organizers: Bernd Sturmfels (MPI Leipzig) and Lauren Williams (Harvard).

In recent years, ideas from integrable systems and scattering amplitudes have led to advances in nonlinear algebra and combinatorics. In this short workshop, aimed at younger participants in the field, we will explore some of the interactions between the above topics.

The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA. For a list of lodging options convenient to the Center, please visit our recommended lodgings page.

All non-Harvard affiliated visitors to the CMSA building are required to complete this covid form prior to arrival: https://forms.gle/xKykcNcXq7ciZuvJ8

Abstract: Matroid theory provides a combinatorial model for linearity, but it plays useful roles beyond linearity. In the classical setup, a linear subspace V of an n-dimensional vector space gives rise to a matroid M(V) on {1,…,n}. However, the matroid M(V) also knows about some nonlinear geometric spaces related to V. Conversely, those nonlinear spaces teach us things we didn’t know about matroids. My talk will discuss some examples.

10:30 am–11:00 am

COFFEE BREAK

11:00 am–11:45 am

Chris Eur

Title: Tautological classes of matroids

Abstract: Algebraic geometry has furnished fruitful tools for studying matroids, which are combinatorial abstractions of hyperplane arrangements. We first survey some recent developments, pointing out how these developments remained partially disjoint. We then introduce certain vector bundles (K-classes) on permutohedral varieties, which we call “tautological bundles (classes)” of matroids, as a new framework that unifies, recovers, and extends these recent developments. Our framework leads to new questions that further probe the boundary between combinatorics and geometry. Joint work with Andrew Berget, Hunter Spink, and Dennis Tseng.

11:45 am–2:00 pm

LUNCH BREAK

2:00 pm–2:45 pm

Nick Early

Title: Biadjoint Scalars and Associahedra from Residues of Generalized Amplitudes

Abstract: The associahedron is known to encapsulate physical properties such as the notion of tree-level factorization for one of the simplest Quantum Field Theories, the biadjoint scalar, which has only cubic interactions. I will discuss novel instances of the associahedron and the biadjoint scalar in a class of generalized amplitudes, discovered by Cachazo, Early, Guevara and Mizera, by taking certain limits thereof. This connection leads to a simple proof of a new realization of the associahedron involving a Minkowski sum of certain positroid polytopes in the second hypersimplex.

2:45 pm–3:30 pm

Anna Seigal

Title: Invariant theory for maximum likelihood estimation

Abstract: I will talk about work to uncover connections between invariant theory and maximum likelihood estimation. I will describe how norm minimization over a torus orbit is equivalent to maximum likelihood estimation in log-linear models. We will see the role played by polytopes and discuss connections to scaling algorithms. Based on joint work with Carlos Améndola, Kathlén Kohn, and Philipp Reichenbach.

3:30 pm–4:00 pm

COFFEE BREAK

4:00 pm–4:45 pm

Matteo Parisi

Title: Amplituhedra, Scattering Amplitudes, and Triangulations

Abstract: In this talk I will discuss about Amplituhedra – generalizations of polytopes inside the Grassmannian – introduced by physicists to encode interactions of elementary particles in certain Quantum Field Theories. In particular, I will explain how the problem of finding triangulations of Amplituhedra is connected to computing scattering amplitudes of N=4 super Yang-Mills theory. Triangulations of polygons are encoded in the associahedron, studied by Stasheff in the sixties; in the case of polytopes, triangulations are captured by secondary polytopes, constructed by Gelfand et al. in the nineties. Whereas a “secondary” geometry describing triangulations of Amplituhedra is still not known, and we pave the way for such studies. I will discuss how the combinatorics of triangulations interplays with T-duality from String Theory, in connection with the Momentum Amplituhedron. A generalization of T-duality led us to discover a striking duality between Amplituhedra of “m=2” type and a seemingly unrelated object – the Hypersimplex. The latter is a polytope which appears in many contexts, from matroid theory to tropical geometry. Based on joint works with Lauren Williams, Melissa Sherman-Bennett, Tomasz Lukowski.

4:45 pm–5:30 pm

Melissa Sherman-Bennett

Title: The hypersimplex and the m=2 amplituhedron

Abstract: In this talk, I’ll continue where Matteo left off. I’ll give some more details about the curious correspondence between the m=2 amplituhedron, a 2k-dimensional subset of Gr(k, k+2), and the hypersimplex, an (n-1)-dimensional polytope in R^n. The amplituhedron and hypersimplex are both images of the totally nonnegative Grassmannian under some map (the amplituhedron map and the moment map, respectively), but are different dimensions and live in very different ambient spaces. I’ll talk about joint work with Matteo Parisi and Lauren Williams in which we give a bijection between decompositions of the amplituhedron and decompositions of the hypersimplex (originally conjectured by Lukowski–Parisi–Williams). The hypersimplex decompositions are closely related to matroidal subdivisions. Along the way, we prove a nice description of the m=2 amplituhedron conjectured by Arkani-Hamed–Thomas–Trnka and give a new decomposition of the m=2 amplituhedron into Eulerian-number-many chambers, inspired by an analogous triangulation of the hypersimplex into Eulerian-number-many simplices.

Thursday, April 28, 2022

9:30 am–10:30 am

Claudia Fevola

Title: Nonlinear Algebra meets Feynman integrals

Abstract: Feynman integrals play a central role in particle physics in the theory of scattering amplitudes. They form a finite-dimensional vector space and the elements of a basis are named “master integrals” in the physics literature. The number of master integrals has been interpreted in different ways: it equals the dimension of a twisted de Rham cohomology group, the Euler characteristic of a very affine variety, and the holonomic rank of a D-module. In this talk, we are interested in a more general family of integrals that contains Feynman integrals as a special case. We explore this setting using tools coming from nonlinear algebra. This is an ongoing project with Daniele Agostini, Anna-Laura Sattelberger, and Simon Telen.

10:30 am–11:00 am

COFFEE BREAK

11:00 am–11:45 am

Simon Telen

Title: Landau discriminants

Abstract: The Landau discriminant is a projective variety containing kinematic parameters for which a Feynman integral can have singularities. We present a definition and geometric properties. We discuss how to compute Landau discriminants using symbolic and numerical methods. Our methods can be used, for instance, to compute the Landau discriminant of the pentabox diagram, which is a degree 12 hypersurface in 6-space. This is joint work with Sebastian Mizera.

11:45 am–2:00 pm

LUNCH BREAK

2:00 pm–2:45 pm

Christian Gaetz

Title: 1-skeleton posets of Bruhat interval polytopes

Abstract: Bruhat interval polytopes are a well-studied class of generalized permutohedra which arise as moment map images of various toric varieties and totally positive spaces in the flag variety. I will describe work in progress in which I study the 1-skeleta of these polytopes, viewed as posets interpolating between weak order and Bruhat order. In many cases these posets are lattices and the polytopes, despite not being simple, have interesting h-vectors. In a special case, work of Williams shows that Bruhat interval polytopes are isomorphic to bridge polytopes, so that chains in the 1-skeleton poset correspond to BCFW-bridge decompositions of plabic graphs.

2:45 pm–3:30 pm

Madeleine Brandt

Title: Top Weight Cohomology of $A_g$

Abstract: I will discuss a recent project in computing the top weight cohomology of the moduli space $A_g$ of principally polarized abelian varieties of dimension $g$ for small values of $g$. This piece of the cohomology is controlled by the combinatorics of the boundary strata of a compactification of $A_g$. Thus, it can be computed combinatorially. This is joint work with Juliette Bruce, Melody Chan, Margarida Melo, Gwyneth Moreland, and Corey Wolfe.

3:30 pm–4:00 pm

COFFEE BREAK

4:00 pm–5:00 pm

Emma Previato

Title: Sigma function on curves with non-symmetric semigroup

Abstract: We start with an overview of the correspondence between spectral curves and commutative rings of differential operators, integrable hierarchies of non-linear PDEs and Jacobian vector fields. The coefficients of the operators can be written explicitly in terms of the Kleinian sigma function: Weierstrass’ sigma function was generalized to genus greater than one by Klein, and is a ubiquitous tool in integrability. The most accessible case is the sigma function of telescopic curves. In joint work with J. Komeda and S. Matsutani, we construct a curve with non-symmetric Weierstrass semigroup (equivalently, Young tableau), consequently non-telescopic, and its sigma function. We conclude with possible applications to commutative rings of differential operators.

Abstract: It is well-known that soliton solutions of the KdV hierarchy are obtained by singular limits of hyper-elliptic curves. However, there is no general results for soliton solutions of the KP hierarchy, KP solitons. In this talk, I will show that some of the KP solitons are related to the singular space curves associated with certain class of numerical semigroups.

10:00 am–10:30 am

COFFEE BREAK

10:30 am–11:15 am

Yelena Mandelshtam

Title: Curves, degenerations, and Hirota varieties

Abstract: The Kadomtsev-Petviashvili (KP) equation is a differential equation whose study yields interesting connections between integrable systems and algebraic geometry. In this talk I will discuss solutions to the KP equation whose underlying algebraic curves undergo tropical degenerations. In these cases, Riemann’s theta function becomes a finite exponential sum that is supported on a Delaunay polytope. I will introduce the Hirota variety which parametrizes all KP solutions arising from such a sum. I will then discuss a special case, studying the Hirota variety of a rational nodal curve. Of particular interest is an irreducible subvariety that is the image of a parameterization map. Proving that this is a component of the Hirota variety entails solving a weak Schottky problem for rational nodal curves. This talk is based on joint work with Daniele Agostini, Claudia Fevola, and Bernd Sturmfels.

11:15 am–12:00 pm

Charles Wang

Title: Differential Algebra of Commuting Operators

Abstract: In this talk, we will give an overview of the problem of finding the centralizer of a fixed differential operator in a ring of differential operators, along with connections to integrable hierarchies and soliton solutions to e.g. the KdV or KP equations. Given these interesting connections, it is important to be able to compute centralizers of differential operators, and we discuss how to use techniques from differential algebra to approach this question, as well as how having these computational tools can help in understanding the structure of soliton solutions to these equations.

12:00 pm–2:00 pm

LUNCH BREAK

2:00 pm–3:00 pm

Sebastian Mizera

Title: Feynman Polytopes

Abstract: I will give an introduction to a class of polytopes that recently emerged in the study of scattering amplitudes in quantum field theory.

3:00 pm–3:30 pm

COFFEE BREAK

3:30 pm–4:30 pm

Nima Arkani-Hamed

Title: Spacetime, Quantum Mechanics and Combinatorial Geometries at Infinity

In recent years, ideas from integrable systems and scattering amplitudes have led to advances in nonlinear algebra and combinatorics. In this short workshop, aimed at younger participants in the field, we will explore some of the interactions between the above topics.

Invited speakers include Federico Ardila, Nima Arkani-Hamed, Madeleine Brandt, Freddy Cachazo, Chris Eur, Claudia Fevola, Christian Gaetz, Yuji Kodama, Fatemeh Mohammadi, Matteo Parisi, Anna Seigal, Melissa Sherman-Bennett, Simon Telen, and Charles Wang.

The workshop will be held in room G10 of the CMSA, located at 20 Garden Street, Cambridge, MA

This event has been cancelled for today, April 29, 2022.

I will talk about joint work with Andrew Lobb related to Toeplitz’s square peg problem, which asks whether every (continuous) Jordan curve in the Euclidean plane contains the vertices of a square. Specifically, we show that every smooth Jordan curve contains the vertices of a cyclic quadrilateral of any similarity class. I will describe the context for the result and its proof, which involves symplectic geometry in a surprising way.

This seminar was rescheduled from April 15, 2022**