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Current Developments in Mathematics 2024
April 5, 2024 - April 6, 2024     
Current Developments in Mathematics 2024 April 5-6, 2024 Harvard University Science Center Lecture Hall C Register Here   Speakers: Daniel Cristofaro-Gardiner - University of Maryland...
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  • CMSA EVENT: CMSA General Relativity Seminar: Noncompact n-dimensional Einstein spaces as attractors for the Einstein flow

    Speaker: Jinhua Wang – Xiamen University

    10:00 AM-11:00 AM
    February 6, 2024
    We prove that along with the Einstein flow, any small perturbations of an $n$($n\geq4$)-dimensional, non-compact negative Einstein space with some “non-positive Weyl tensor” lead to a unique and global solution, and the solution will be attracted to a noncompact Einstein space that is close to the background one. The $n=3$ case has been addressed by Wang-Yuan, while in dimension $n\geq 4$, as we know, negative Einstein metrics in general have non-trivial moduli spaces. This fact is reflected on the structure of Einstein equations, which further indicates no decay for the spatial Weyl tensor. Furthermore, it is suggested in the proof that the mechanic preventing the metric from flowing back to the original Einstein metric lies in the non-decaying character of spatial Weyl tensor. In contrary to the compact case considered in Andersson-Moncrief, our proof is independent of the theory of infinitesimal Einstein deformations. Instead, we take advantage of the inherent geometric structures of Einstein equations and develop an approach of energy estimates for a hyperbolic system of Maxwell type.
    Please note: This seminar will take place on Zoom from 10:00 am to 11:00 am ET


    Password: cmsa

  • SEMINARS: Probability Seminar: Fractal Geometry of Stochastic Partial Differential Equations

    Speaker: Promit Ghosal – Brandeis

    1:30 PM-2:30 PM
    February 6, 2024

    Stochastic partial differential equations (PDEs) find extensive applications across diverse domains such as physics, finance, biology, and engineering, serving as effective tools for modeling systems influenced by random factors. The analysis of the patterns in the peaks and valleys of stochastic PDEs is crucial for gaining deeper insights into the underlying physical phenomena.

    One notable example is the KPZ equation, a fundamental stochastic PDE associated with significant models like random growth processes, Burgers turbulence, interacting particle systems, and random polymers. The study of the fractal structures inherent in the KPZ equation provides a quantitative characterization of the intermittent nature of its peaks, as well as those of the stochastic heat equation—a subject that has been extensively explored over the past few decades.

    Conversely, the Parabolic Anderson model (PAM) serves as a prototypical framework for simulating the conduction of electrons in crystals containing defects. Investigating the intermittency of peaks in the PAM has been a prominent area of research, closely tied to the phenomenon of Anderson localization.

    In this presentation, we delve into the fractal geometry of both the KPZ equation and the PAM, unveiling their multifractal nature. Specifically, we demonstrate that the spatial and spatio-temporal peaks of these equations exhibit infinitely many distinct values. Furthermore, we compute the macroscopic Hausdorff dimension (introduced by Barlow and Taylor) associated with these peaks.

    The key findings presented here stem from a series of works that employ a diverse array of tools, ranging from random matrix theory and the Gibbs property of random curves to the utilization of regularity structures and paracontrolled calculus.

  • HARVARD-MIT ALGEBRAIC GEOMETRY SEMINAR: Harvard-MIT Algebraic Geometry Seminar: Enumerativity of fixed-domain Gromov-Witten invariants

    Speaker: Carl Lian – Tufts University

    3:00 PM-4:00 PM
    February 6, 2024

    It is well-understood that Gromov-Witten (GW) invariants often fail to be enumerative. For example, when r is at least 3, the higher-genus GW invariants of P^r fail to count smooth curves in projective space in any transparent sense. The situation seems to be better when one fixes the complex structure of the domain curve. It was originally speculated that if X is a Fano variety, then the “fixed-domain” GW count of curves of sufficiently large degree passing through the maximal number of general points is enumerative. I will discuss some positive and negative results in this direction, focusing on the case of hypersurfaces. The most recent results are joint with Roya Beheshti, Brian Lehmann, Eric Riedl, Jason Starr, and Sho Tanimoto, and build on earlier work with Rahul Pandharipande and Alessio Cela.


    For more information, please see

  • CMSA EVENT: CMSA/Tsinghua Math-Science Literature Lecture: Stretching and shrinking: 85 years of the Hopf argument for ergodicity

    Speaker: Amie Wilkinson – University of Chicago

    9:00 AM-10:30 AM
    February 7, 2024
    The early 20th century witnessed an explosion of activity, much of it centered at Harvard, on rigorizing the property of ergodicity first proposed by Boltzmann in his 1898  Ergodic Hypothesis for ideal gases. Earlier, in the 1880’s, Henri Poincaré and Felix Klein had also initiated a study of discrete groups of hyperbolic isometries. The geodesics in hyperbolic manifolds were discovered to carry a rich structure, first investigated from a topological perspective by Emil Artin and Marston Morse.  The time was ripe to investigate geodesics in hyperbolic manifolds from an ergodic theoretic (i.e., statistical) perspective, and indeed Gustav Hedlund proved in 1934 that the geodesic flow for closed hyperbolic surfaces is ergodic.
    In 1939, Eberhard Hopf published a proof of the ergodicity of geodesic flows for negatively curved surfaces containing a novel method, now known as the Hopf argument.  The Hopf argument, a “soft” argument for ergodicity of systems with some hyperbolicity (the “stretching and shrinking” in the title) has since seen wide application in geometry, representation theory and dynamics.  I will discuss three results relying on the Hopf argument:
    Theorem (E. Hopf, 1939, D. Anosov, 1967): In a closed manifold of negative sectional curvatures, almost every geodesic is directionally equidistributed.
    Theorem (G. Mostow, 1968) Let M and N be closed hyperbolic manifolds of dimension at least 3, and let f:M->N be a homotopy equivalence.  Then f is homotopic to a unique isometry.
    Theorem (R. Mañé, 1983, A. Avila- S. Crovisier- A.W., 2022) The C^1 generic symplectomorphism of a closed symplectic manifold with positive entropy is ergodic.
    Register here to attend virtually:  Zoom Webinar Registration
  • CMSA EVENT: CMSA New Technologies in Mathematics Seminar: Large language models, mathematical discovery, and search in the space of strategies: an anecdote

    Speaker: Jordan Ellenberg – Univ. of Wisconsin Dept. of Mathematics

    1:00 PM-2:00 PM
    February 7, 2024
    20 Garden Street, Cambridge, MA 02138

    Please note special time

    I spent a portion of 2023 working with a team at DeepMind on the “cap set problem” – how large can a subset of (Z/3Z)^n be which contains no three terms which sum to zero? (I will explain, for those not familiar with this problem, something about the role it plays in combinatorics, its history, and why number theorists care about it a lot.) By now, there are many examples of machine learning mechanisms being used to help generate interesting mathematical knowledge, and especially interesting examples. This project used a novel protocol; instead of searching directly for large cap sets, we used LLMs trained on code to search the space of short programs for those which, when executed, output large capsets. One advantage is that a program is much more human-readable than a large collection of vectors over Z/3Z, bringing us closer to the not-very-well-defined-but-important goal of “interpretable machine learning.” I’ll talk about what succeeded in this project (more than I expected!) what didn’t, and what role I can imagine this approach to the math-ML interface playing in near-future mathematical practice.

    The paper:
    Password: cmsa

  • HARVARD-MIT COMBINATORICS SEMINAR: Richard P. Stanley Seminar in Combinatorics: Cluster algebras and scattering amplitudes

    Speaker: Marcus Spradlin – Brown University

    4:15 PM-5:15 PM
    February 7, 2024

    In recent years fruitful connections between math and physics have emerged from the study of scattering amplitudes. I will review and put into context some key concepts from this exchange of ideas, involving cluster algebras, positive geometries, and the amplituhedron. I will highlight further physics-motivated conjectures that may provide fruitful avenues for continued exchange in the years to come.


    For more info, see

  • OPEN NEIGHBORHOOD SEMINAR: Open Neighborhood Seminar: Symmetry in Deep Neural Networks

    Speaker: Robin Walters – Northeastern

    4:30 PM-5:30 PM
    February 7, 2024
    1 Oxford Street, Cambridge, MA 02138 USA

    Deep learning has had transformative impacts in many fields including computer vision, computational biology, and dynamics by allowing us to learn functions directly from data. However, there remain many domains in which learning is difficult due to poor model generalization or limited training data. We’ll explore two applications of representation theory to neural networks which help address these issues. Firstly, consider the case in which the data represent a group equivariant function. In this case, we can consider spaces of equivariant neural networks which may more easily be fit to the data using gradient descent. Secondly, we can consider symmetries of the parameter space as well. Exploiting these symmetries can lead to models with fewer free parameters, faster convergence, and more stable optimization.



  • CMSA EVENT: CMSA Colloquium: Machine learning and scientific computing: there is plenty of room in the middle

    Speaker: Petros Koumoutsakos – Harvard SEAS

    4:30 PM-5:30 PM
    February 12, 2024
    20 Garden Street, Cambridge, MA 02138

    Over the last last thirty years we have experienced more than a billion-fold increase in hardware capabilities and a dizzying pace of acquiring and transmitting massive amounts of data. Scientific Computing and, more lately, Artificial Intelligence (AI) has been key beneficiaries of these advances. In this talk I would outline the need for bridging the decades long advances in Scientific Computing with those of AI. I will use examples from fluid mechanics to argue for forming alloys of AI and simulations for their prediction and control. I will present novel algorithms for learning the Effective Dynamics (LED) of complex systems and a fusion of multi- agent reinforcement learning and scientific computing (SciMARL) for modeling and control of turbulent flows. I will also show our recent work on Optimizing a Discrete Loss (ODIL) that outperforms popular techniques such as PINNs by several orders of magnitude.
    I will juxtapose successes and failures and argue that the proper fusion of scientific computing and AI expertise are essential to advance scientific frontiers.

  • CMSA EVENT: CMSA General Relativity Seminar: Characteristic Initial Value Problem for the 3D Compressible Euler Equations

    Speaker: Sifan Yu – NUS

    11:00 AM-12:00 PM
    February 13, 2024
    20 Garden Street, Cambridge, MA 02138
    We present the first result for the characteristic initial value problem of the compressible Euler equations in three space dimensions without any symmetry assumption. We allow presence of vorticity and consider any equation of state. Compared to the standard Cauchy problem, where initial data can be freely prescribed on a constant-time hypersurface, we formulate the problem by distinguishing between the “free-component” and the “constrained-component” of the initial data. The latter is to be solved by the “free-component” utilizing the properties of the compressible Euler equations on the initial null hypersurfaces. Then, we establish a priori estimates, followed by a local well-posedness and a continuation criterion argument. Moreover, we prove a regularity theory in Sobolev norms. Our analysis critically relies on the vectorfield method due to the nature of the problem. This is a joint work with Jared Speck.


    Password: cmsa

  • SEMINARS: Probability Seminar: POSTPONED

    Speaker: Matthew Nicoletti – MIT

    1:30 PM-2:30 PM
    February 13, 2024

    This seminar has been POSTPONED. Apologies for any inconveniences. Rescheduled date TBD.

    Recently, there has been much progress in understanding stationary measures for colored (also called multi-species or multi-type) interacting particle systems, motivated by asymptotic phenomena and rich underlying algebraic and combinatorial structures (such as nonsymmetric Macdonald polynomials).

    In this work, we present a unified approach to constructing stationary measures for several colored particle systems on the ring and the line, including (1) the Asymmetric Simple Exclusion Process (mASEP); (2) the q-deformed Totally Asymmetric Zero Range Process (TAZRP) also known as the q-Boson particle system; (3) the q-deformed Pushing Totally Asymmetric Simple Exclusion Process (q-PushTASEP). Our method is based on integrable stochastic vertex models and the Yang–Baxter equation. We express the stationary measures as partition functions of new “queue vertex models” on the cylinder. The stationarity property is a direct consequence of the Yang–Baxter equation. This is joint work with A. Aggarwal and L. Petrov.


    Harvard-MIT Algebraic Geometry Seminar: POSTPONED

    Speaker: Maksym Fedorchuk – Boston College

    3:00 PM-4:00 PM
    February 13, 2024
    Seminar POSTPONED due to weather. Apologies for any inconveniences.
    Rescheduled date: February 27th, 3:00-4:00. See website for more details.
    A recent achievement in K-stability of Fano varieties is an algebro-geometric construction of a projective moduli space of K-polystable Fanos. The ample line bundle on this moduli space is the CM line bundle of Tian. One of the consequences of the general theory is that given a family of K-stable Fanos over a punctured curve, the polystable filling is the one that minimizes the degree of the CM line bundle after every finite base change. A natural question is to ask what are the CM-minimizers without base change. In answering this question, we arrive at a theory of Kollár stability for fibrations over one-dimensional bases, and standard models of Fano fibrations. After explaining the general theory, I will sketch work in progress on standard models of quartic threefold hypersurfaces. This talk is based on joint work with Hamid Abban and Igor Krylov.

    For more information, please see

  • THURSDAY SEMINAR SEMINAR: Thursday Seminar: Cyclotomic redshift

    Thursday Seminar: Cyclotomic redshift

    Speaker: Andy Senger – Harvard

    3:30 PM-5:30 PM
    February 15, 2024
    1 Oxford Street, Cambridge, MA 02138 USA

    I will discuss the interaction of telescopically localized algebraic K theory with the higher cyclotomic extensions introduced in Mike’s talk, and explain why this is a key step in the disproof of the telescope conjecture. Along the way, we will show that telescopically localized K theory commutes with (co)limits indexed by pi finite p-spaces.

  • SEMINARS: Probability Seminar: Optimal rigidity and maximum of the characteristic polynomial of Wigner matrices

    Speaker: Patrick Lopatto – Brown University

    1:30 PM-2:30 PM
    February 20, 2024

    We consider two related questions about the extremal statistics of Wigner matrices (random symmetric matrices with independent entries). First, how much can their eigenvalues fluctuate? It is known that the eigenvalues of such matrices display repulsive interactions, which confine them near deterministic locations. We provide optimal estimates for this “rigidity” phenomenon. Second, what is the behavior of the maximum of the characteristic polynomial? This is motivated by a conjecture of Fyodorov-Hiary-Keating on the maxima of logarithmically correlated fields, and we will present the first results on this question for Wigner matrices. This talk is based on joint work with Paul Bourgade and Ofer Zeitouni.

  • HARVARD-MIT ALGEBRAIC GEOMETRY SEMINAR: Harvard-MIT Algebraic Geometry Seminar: Brill-Noether loci

    Harvard-MIT Algebraic Geometry Seminar: Brill-Noether loci

    Speaker: Montserrat Teixidor – Tufts University

    3:00 PM-4:00 PM
    February 20, 2024
    1 Oxford Street, Cambridge, MA 02138 USA
    Brill-Noether loci are defined as the set of curves of genus g that have an unexpected linear series of degree d and dimension r.

    Pflueger showed that these loci are non-empty when the expected codimension is at most g-3. By studying linear series on chains of elliptic curves, we give a new proof of a slightly refined version of this result. We can also look at the behavior of the generic curve in the locus.

    An interesting conjecture of Auel and Haburcak states that these loci are distinct and not contained in each other, unless they come from adding or removing fixed points. Their proof made use of curves contained in K3 surfaces and was sufficient to prove the result in small genus. Using chains of elliptic curves, we can obtain additional information.


    For more information, please see

  • CMSA EVENT: Math Science Lectures in Honor of Raoul Bott: Maggie Miller: Fibered ribbon knots vs. major 4D conjectures, Lecture 1

    Speaker: Maggie Miller – University of Texas at Austin

    4:00 PM-5:30 PM
    February 20, 2024
    1 Oxford Street, Cambridge, MA 02138

    View from the CMSA Events Page

    Fibered ribbon knots vs. major 4D conjectures

    Location: Harvard University Science Center Hall A & via Zoom webinar

    Dates: Feb 20 & 22, 2024

    Time: 4:00-5:30 pm

    Directions and Recommended Lodging

    Registration is required.

    Maggie Miller is an assistant professor in the mathematics department at the University of Texas at Austin and a Clay Research Fellow.

    This will be the fourth annual Math Science Lecture Series held in Honor of Raoul Bott.

    Fibered ribbon knots vs. major 4D conjectures

    Feb. 20, 2024

    Title: Fibered ribbon knots and the Poincaré conjecture

    Abstract: A knot is “fibered” if its complement in S^3 is the total space of a bundle over the circle, and ribbon if it bounds a smooth disk into B^4 with no local maxima with respect to radial height. A theorem of Casson-Gordon from 1983 implies that if a fibered ribbon knot does not bound any fibered disk in B^4, then the smooth 4D Poincaré conjecture is false. I’ll show that unfortunately (?) many ribbon disks bounded by fibered knots are fibered, giving some criteria for extending fibrations and discuss how one might search for non-fibered examples.


    Feb. 22, 2024

    Title: Fibered knots and the slice-ribbon conjecture

    Abstract: The slice-ribbon conjecture (Fox, 1962) posits that if a knot bounds any smooth disk into B^4, it also bounds a ribbon disk. The previously discussed work of Casson-Gordon yields an obstruction to many fibered knots being ribbon, yielding many interesting potential counterexamples to this conjecture — if any happy to bound a non-ribbon disk. In 2022, Dai-Kong-Mallick-Park-Stoffregen showed that unfortunately (?) many of these knots don’t bound a smooth disk into B^4 and thus can’t disprove the conjecture. I’ll show a simple alternate proof that a certain interesting knot (the (2,1)-cable of the figure eight) isn’t slice and discuss remaining open questions. This talk is joint with Paolo Aceto, Nickolas Castro, JungHwan Park, and Andras Stipsicz.

    Talk Chair: Cliff Taubes (Harvard Mathematics)

    Moderator: Freid Tong (Harvard CMSA)

    Raoul Bott (9/24/1923 – 12/20/2005) is known for the Bott periodicity theorem, the Morse–Bott functions, and the Borel–Bott–Weil theorem. For more info, please see the article “Remembering Raoul Bott”  from the American Mathematical Society.

  • CMSA EVENT: CMSA Member Seminar: Integrability and Hidden Symmetries in Black Hole Dynamics

    Speaker: Uri Kol – Harvard

    12:00 PM-1:00 PM
    February 23, 2024

    The last decade has produced a number of remarkable discoveries, such as the first direct observation of gravitational waves by the LIGO/Virgo collaboration and the first black hole image taken by the Event Horizon Telescope. These discoveries mark the beginning of a new precision era in black hole physics, which is expected to develop further by future experiments such as LISA, the Einstein Telescope and Cosmic Explorer.

    In the era of precision black hole measurements, there is a need for precision theoretical methods and accurate predictions. In this talk I will describe an integrable sector of the gravitational scattering problem – analogous to the hydrogen atom in quantum mechanics – in which exact predictions can be made, and the implications for astrophysical black holes and binary mergers.

    Friday, Feb. 23rd at 12pm, with lunch, lounge at CMSA (20 Garden Street). Also by Zoom:

  • HARVARD-MIT COMBINATORICS SEMINAR: Richard P. Stanley Seminar in Combinatorics: Asymptotic separation index as a tool in descriptive combinatorics

    Speaker: Anton Bernshteyn – Georgia Tech

    3:00 PM-4:00 PM
    February 23, 2024

    A common theme throughout mathematics is the search for “constructive” solutions to problems as opposed to mere existence results. For problems over R and other well-behaved spaces, this idea is nicely captured by the concept of a Borel construction. In particular, one can investigate Borel solutions to classical combinatorial problems such as graph colorings, perfect matchings, etc. The area studying these questions is called descriptive combinatorics. As I will explain in the talk, many facts in graph theory that we know and love—for example, Brooks’ theorem—turn out to be inherently “non-constructive” in this sense. The main result of this talk is that Borel versions of various classical combinatorial theorems nevertheless hold on graphs that can, in some sense, be easily decomposed into subgraphs with finite components. No prior familiarity with Borel combinatorics or descriptive set theory will be assumed. Based on joint work with Felix Weilacher.


    For more info, see

  • GAUGE-TOPOLOGY-SYMPLECTIC SEMINAR: Gauge Theory and Topology Seminar: Spectral flow and reducible solutions to the massive Vafa-Witten equations

    Speaker: Cliff Taubes – Harvard University

    3:30 PM-4:30 PM
    February 23, 2024
    1 Oxford Street, Cambridge, MA 02138 USA

    The Vafa-Witten equations (with or without a mass term) constitute a non-linear, first order system of differential equations on a given oriented, compact, Riemannian 4-manifold. Because these are the variational equations of a functional, the linearized equations at any given solution can be used to define an elliptic, first order, self-adjoint differential operator. This talk will describe bounds (upper and lower) for the spectral flow between respective versions of this operator that are defined by the elements in diverging sequences of reducible solutions. (The spectral flow is formally the difference between the respective Morse indices of the solutions when they are viewed as critical points of the functional.) In some cases, the absolute value of the spectral flow is bounded along the sequence, whereas in others it diverges. This is a curious state of affairs.