Joint with Moritz Kerz. We show a structure theorem on those loci which enables us to deduce several consequences, one of which being Hard Lefschetz in rank one in positive characteristic.

The Gauss curvature flow was introduced to model the shape of worn stones. It is a parabolic Monge-Ampere type flow for convex hypersurfaces, involving regularity issues for fully nonlinear equations and free boundary problems, and involving several entropy. In this talk, we will discuss the free boundary problem concerning worn stone with flat sides, and the optimal regularity of non-concave fully nonlinear equations. When time permits, we also talk about the singularity analysis, especially about the relations to the affine geometry at the critical case, to fully nonlinear equations at the super-critical case, and to convex geometry and spectral analysis at the sub-critical case.

After reviewing the definition of intrinsic flat convergence, I will present open problems on the almost rigidity or stability of classic rigidity theorems for manifolds with scalar curvature bounds and survey partial results towards these conjectures.

The absolute cohomological purity for étale cohomology of Gabber–Thomasson implies that an étale cohomology class on a regular scheme extends uniquely over a closed subscheme of large codimension. I will discuss the corresponding phenomenon for flat cohomology. The talk is based on joint work with Peter Scholze.

We present two concrete examples where the Renyi rather than just the von Neumann entanglement entropy is necessary in order to obtain certain insights into quantum many-body systems.

In the first example, we consider systems supporting ballistic information propagation and diffusive transport. It is well known that the linear-in-time growth of the von Neumann entanglement entropy (starting from a product state) is a probe of the former. Perhaps surprisingly, we show that the Renyi entanglement entropy (with Renyi index greater than 1) grows diffusively (i.e., as a square root of time) and is consequently a probe of the latter.

In the second example, we study the problem of approximating local properties of a quantum many-body state using matrix product and projected entangled pair representations in one and two dimensions, respectively. We prove that area laws for the Renyi entanglement entropy (with Renyi index less than 1) lead to nontrivial upper bounds on the bond dimension. The bounds only depend on the accuracy of the desired approximation but not the system size.

The Strominger-Yau-Zaslow conjecture predicts the existence of special Lagrangians fibration in Calabi-Yau manifolds near large complex structure limit. The SYZ conjecture has been an important guiding principle for mirror symmetry and many of the implications are verified. In this talk, I will report on the recent progress on the SYZ fibration on certain log Calabi-Yau surfaces using the Lagrangian mean curvature flow and the theory of J-holomorphic curves. As a bi-product, we produce many new special Lagrangian submanifolds. I will also explain the applications in mirror symmetry, including the tropical/holomorphic correspondence for log Calabi-Yau surfaces and a mathematical realization of renormalization process of Hori-Vafa. Part of the talk is based on the joint work with T. Collins, A. Jacob and S-C. Lau, T-J. Lee.

Consider these basic questions: What can we say about finite sums of powers of consecutive whole numbers? What can we say about whole number solutions to polynomial equations? What about factorizations into primes? What about values of the Riemann zeta function? In interesting families of examples — elementary and sophisticated, ancient and modern — “Bernoulli numbers” unify these seemingly unrelated questions. After an introduction to the Bernoulli numbers, we will explore related developments for these intertwined problems, which lead to central challenges in number theory and beyond.

In the mid-1800s, Kummer observed some striking congruences between certain values of the Riemann zeta function, which have important consequences in algebraic number theory, in particular for unique factorization in certain rings. In spite of its potential, this topic lay mostly dormant for nearly a century until it was revived by Iwasawa in the mid-1950s. Since then, advances in arithmetic geometry and number theory (in particular, for modular forms, certain analytic functions that play a central role in number theory) have enabled substantial extension to congruences in the context of other arithmetically significant data, and this has remained an active area of research. In this talk, I will survey old and new tools for studying such congruences. I will conclude by introducing some unexpected challenges that arise when one tries to take what would seem like immediate next steps beyond the current state of the art.

Will a mosquito survive raindrop collisions? How the bubbles under a ship reduce the drag force? In nature and industry, flows with drops and bubbles exist everywhere. To understand these flows, one of the powerful tools is the direct numerical simulation (DNS). Among all the DNS methods, we choose the Phase Field (PF) method and develop some models based on it to simulate the complicated flows, such as flows with moving contact lines, fluid-structure interaction, ternary fluids and turbulence. In this talk, I will firstly introduce the advantages and disadvantages of PF method. Then, I will show its applications: drop impact on an object, compound droplet dynamics, water entry of an object and multiphase turbulence.

In a recent joint work with Kisin and Shin, we compared the Grothendieck-Lefschetz trace formula for abelian-type Shimura varieties with the stable Arthur-Selberg trace formulas. In this talk I will sketch some key ideas in the proof.

Let $s_n(M_n)$ denote the smallest singular value of an $n\times n$ random matrix $M_n$. We will discuss a novel combinatorial approach (in particular, not using either inverse Littlewood–Offord theory or net arguments) for obtaining upper bounds on the probability that $s_n(M_n)$ is smaller than $\eta \geq 0$ for quite general random matrix models. Such estimates are a fundamental part of the non-asymptotic theory of random matrices and have applications to the strong circular law, numerical linear algebra etc. In several cases of interest, our approach provides stronger bounds than those obtained by Tao and Vu using inverse Littlewood–Offord theory.

For Rayleigh-Bénard under geometrical confinement, under rotation or the double diffusive convection with the second scalar component stabilizing the convective flow, they seem to be the three different canonical models in turbulent flow. However, previous research coincidentally reported the scalar transport enhancement in these systems. The results are counter-intuitive because the higher efficiency of scalar transport is bought about by the slower flow. In this talk, I will show you a fundamental and unified perspective on such the global transport behavior observed in the seemingly different systems. We further show that the same view can be applied to the quasi-static magnetoconvection, and indeed the regime with heat transport enhancement has been found. The beauty of physics is to understand the seemingly unrelated phenomena by a simplified concept. Here we provide a simplified and generic view, and this concept could be potentially extended to other situations where the turbulent flow is subjected to an additional stabilization.

I will discuss non-perturbative definitions of quantum field theories, some properties of correlation functions of local operators, and give a brief overview of some results and open questions concerning the conformal bootstrap.

In his landmark paper of 1900 in which he derived his law of blackbody radiation from Boltzmann’s theory of entropy, Planck invoked as evidence inferred values for a series of theoretically interlinked microphysical constants none of which had been measured with precision before. In doing so he initiated a kind of evidence that had never been successfully deployed before, yet quickly came to be central to modern microphysics. The talk will examine the distinctive logic underlying this kind of evidence and sketch the history of its becoming increasingly authoritative before ending with questions first about just what it is evidence for and finally about the extent to which the measurements achieving extraordinarily precise values of the constants can legitimately be said to have gained experimental access to specifics of the microphysical realm.

Will a mosquito survive raindrop collisions? How the bubbles under a ship reduce the drag force? In nature and industry, flows with drops and bubbles exist everywhere. To understand these flows, one of the powerful tools is the direct numerical simulation (DNS). Among all the DNS methods, we choose the Phase Field (PF) method and develop some models based on it to simulate the complicated flows, such as flows with moving contact lines, fluid-structure interaction, ternary fluids and turbulence. In this talk, I will firstly introduce the advantages and disadvantages of PF method. Then, I will show its applications: drop impact on an object, compound droplet dynamics, water entry of an object and multiphase turbulence.

In a recent joint work with Kisin and Shin, we compared the Grothendieck-Lefschetz trace formula for abelian-type Shimura varieties with the stable Arthur-Selberg trace formulas. In this talk I will sketch some key ideas in the proof.

Let $s_n(M_n)$ denote the smallest singular value of an $n\times n$ random matrix $M_n$. We will discuss a novel combinatorial approach (in particular, not using either inverse Littlewood–Offord theory or net arguments) for obtaining upper bounds on the probability that $s_n(M_n)$ is smaller than $\eta \geq 0$ for quite general random matrix models. Such estimates are a fundamental part of the non-asymptotic theory of random matrices and have applications to the strong circular law, numerical linear algebra etc. In several cases of interest, our approach provides stronger bounds than those obtained by Tao and Vu using inverse Littlewood–Offord theory.

For Rayleigh-Bénard under geometrical confinement, under rotation or the double diffusive convection with the second scalar component stabilizing the convective flow, they seem to be the three different canonical models in turbulent flow. However, previous research coincidentally reported the scalar transport enhancement in these systems. The results are counter-intuitive because the higher efficiency of scalar transport is bought about by the slower flow. In this talk, I will show you a fundamental and unified perspective on such the global transport behavior observed in the seemingly different systems. We further show that the same view can be applied to the quasi-static magnetoconvection, and indeed the regime with heat transport enhancement has been found. The beauty of physics is to understand the seemingly unrelated phenomena by a simplified concept. Here we provide a simplified and generic view, and this concept could be potentially extended to other situations where the turbulent flow is subjected to an additional stabilization.

I will discuss non-perturbative definitions of quantum field theories, some properties of correlation functions of local operators, and give a brief overview of some results and open questions concerning the conformal bootstrap.

In his landmark paper of 1900 in which he derived his law of blackbody radiation from Boltzmann’s theory of entropy, Planck invoked as evidence inferred values for a series of theoretically interlinked microphysical constants none of which had been measured with precision before. In doing so he initiated a kind of evidence that had never been successfully deployed before, yet quickly came to be central to modern microphysics. The talk will examine the distinctive logic underlying this kind of evidence and sketch the history of its becoming increasingly authoritative before ending with questions first about just what it is evidence for and finally about the extent to which the measurements achieving extraordinarily precise values of the constants can legitimately be said to have gained experimental access to specifics of the microphysical realm.

Limited funding to help defray travel expenses is available for graduate students and recent PhDs. Please check later for more information. 1. A letter indicating your name, address, current status, university affiliation, citizenship, and area of study. F1 visa holders are eligible to apply for support. 2. If you are a graduate student, please send a brief letter of recommendation from a faculty member to explain the relevance of the conference to your studies or research.

For travel information, click here. Sheraton Commander 16 Garden Street, Cambridge MA 02138 telephone: (617) 547-4800 hotel-Harvard campus: 5-10 minute walk Irving House 24 Irving St., Cambridge MA 02138 telephone: (617) 547-4600, hotel-Harvard campus: 5-10 minute walk Friendly Inn(B + B) 1673 Cambridge St., Cambridge, MA 02138 telephone: (617) 547-7851 hotel-Harvard campus: 5-10 minute walk Courtyard Marriott Cambridge 777 Memorial Drive Cambridge, Massachusetts 02139 USA 617-492-7777, 1-800-321-2211 15-minute walk to Harvard Square, Free shuttle to Harvard Hotel Tria 220 Alewife Brook Pkwy Cambridge, Massachusetts, 02138-1102 (617) 491-8000, outside Harvard Square. 10 minute walk to public transportation Free Shuttle bus to Harvard

“Harvard University is committed to maintaining a safe and healthy educational and work environment in which no member of the University community is, on the basis of sex, sexual orientation, or gender identity, excluded from participation in, denied the benefits of, or subjected to discrimination in any University program or activity. More information can be found here“.

The Event Horizon Telescope image of the supermassive black hole in the galaxy M87 is dominated by a bright, unresolved ring. General relativity predicts that embedded within this image lies a thin “photon ring,” which is composed of an infinite sequence of self-similar subrings that are indexed by the number of photon orbits around the black hole. The subrings approach the edge of the black hole “shadow,” becoming exponentially narrower but weaker with increasing orbit number, with seemingly negligible contributions from high order subrings. In the talk, I will discuss the structure of the photon ring, starting with non-rotating black holes, and then proceeding to the complex patterns that emerge when rotation is taken into account. Subsequently I will argue that the subrings produce strong and universal signatures on long interferometric baselines. These signatures offer the possibility of precise measurements of black hole mass and spin, as well as tests of general relativity, using only a sparse interferometric array.

Limited funding to help defray travel expenses is available for graduate students and recent PhDs. Please check later for more information. 1. A letter indicating your name, address, current status, university affiliation, citizenship, and area of study. F1 visa holders are eligible to apply for support. 2. If you are a graduate student, please send a brief letter of recommendation from a faculty member to explain the relevance of the conference to your studies or research.

For travel information, click here. Sheraton Commander 16 Garden Street, Cambridge MA 02138 telephone: (617) 547-4800 hotel-Harvard campus: 5-10 minute walk Irving House 24 Irving St., Cambridge MA 02138 telephone: (617) 547-4600, hotel-Harvard campus: 5-10 minute walk Friendly Inn(B + B) 1673 Cambridge St., Cambridge, MA 02138 telephone: (617) 547-7851 hotel-Harvard campus: 5-10 minute walk Courtyard Marriott Cambridge 777 Memorial Drive Cambridge, Massachusetts 02139 USA 617-492-7777, 1-800-321-2211 15-minute walk to Harvard Square, Free shuttle to Harvard Hotel Tria 220 Alewife Brook Pkwy Cambridge, Massachusetts, 02138-1102 (617) 491-8000, outside Harvard Square. 10 minute walk to public transportation Free Shuttle bus to Harvard

“Harvard University is committed to maintaining a safe and healthy educational and work environment in which no member of the University community is, on the basis of sex, sexual orientation, or gender identity, excluded from participation in, denied the benefits of, or subjected to discrimination in any University program or activity. More information can be found here“.

The first part of this talk will introduce generalized Jordan–Wigner transformation on arbitrary triangulation of any simply connected manifold in 2d, 3d and general dimensions. This gives a duality between all fermionic systems and a new class of lattice gauge theories. This map preserves the locality and has an explicit dependence on the second Stiefel–Whitney class and a choice of spin structure on the manifold. In the Euclidean picture, this mapping is exactly equivalent to introducing topological terms (Chern-Simon term in 2d or the Steenrod square term in general) to the Euclidean action. We can increase the code distance of this mapping, such that this mapping can correct all 1-qubit and 2-qubits errors and is useful for the simulation of fermions on the quantum computer. The second part of my talk is about SPT phases. By the boson-fermion duality, we are able to show the equivalent between any supercohomology fermionic SPT and some higher-group bosonic SPT phases. Particularly in (3+1)D, we have constructed a unitary quantum circuit for any supercohomology fermionic SPT state with gapped boundary construction. This fermionic SPT state is derived by gauging higher-form symmetry in the higher-group bosonic SPT and ungauging the fermion parity. The bulk-boundary correspondence in (3+1)D fermion SPT phases will also be briefly discussed.

In recent years, para-Hermitian geometry has been used to describe T-duality covariant spacetimes for string theory. In my talk, I will present applications of para-Hermitian geometry to 2D (2,2) SUSY sigma models and show that this geometry gives rise to a new, yet unexplored, notion of mirror symmetry.

Quantum Field Theory is a framework of fundamental physics, which in particular has played important roles in the modern development of various subjects in mathematics, including enumerative geometry, knot theory, and low-dimensional topology. On the other hand, Geometric Representation Theory is a subject in mathematics that studies a linear model of various types of symmetries using powerful techniques of algebraic geometry. In recent years, there has been much progress relating the two subjects, enriching the subject of Geometric Representation Theory. In this colloquium style talk, we will review some recent advancements on the topic. No prior knowledge of either Quantum Field Theory or Geometric Representation Theory will be assumed.

The task of manipulating randomness has been a subject of intense investigation in the theory of computer science. The classical definition of this task consider a single processor massaging random samples from an unknown source and trying to convert it into a sequence of uniform independent bits.

In this talk I will talk about a less studied setting where randomness is distributed among different players who would like to convert this randomness to others forms with relatively little communication. For instance players may be given access to a source of biased correlated bits, and their goal may be to get a common random bit out of this source. Even in the setting where the source is known this can lead to some interesting questions that have been explored since the 70s with striking constructions and some surprisingly hard questions. After giving some background, I will describe a recent work which explores the task of extracting common randomness from correlated sources with bounds on the number of rounds of interaction.

Based on joint works with Mitali Bafna (Harvard), Badih Ghazi (Google) and Noah Golowich (Harvard).

The first part of this talk will introduce generalized Jordan–Wigner transformation on arbitrary triangulation of any simply connected manifold in 2d, 3d and general dimensions. This gives a duality between all fermionic systems and a new class of lattice gauge theories. This map preserves the locality and has an explicit dependence on the second Stiefel–Whitney class and a choice of spin structure on the manifold. In the Euclidean picture, this mapping is exactly equivalent to introducing topological terms (Chern-Simon term in 2d or the Steenrod square term in general) to the Euclidean action. We can increase the code distance of this mapping, such that this mapping can correct all 1-qubit and 2-qubits errors and is useful for the simulation of fermions on the quantum computer. The second part of my talk is about SPT phases. By the boson-fermion duality, we are able to show the equivalent between any supercohomology fermionic SPT and some higher-group bosonic SPT phases. Particularly in (3+1)D, we have constructed a unitary quantum circuit for any supercohomology fermionic SPT state with gapped boundary construction. This fermionic SPT state is derived by gauging higher-form symmetry in the higher-group bosonic SPT and ungauging the fermion parity. The bulk-boundary correspondence in (3+1)D fermion SPT phases will also be briefly discussed.

In recent years, para-Hermitian geometry has been used to describe T-duality covariant spacetimes for string theory. In my talk, I will present applications of para-Hermitian geometry to 2D (2,2) SUSY sigma models and show that this geometry gives rise to a new, yet unexplored, notion of mirror symmetry.

Quantum Field Theory is a framework of fundamental physics, which in particular has played important roles in the modern development of various subjects in mathematics, including enumerative geometry, knot theory, and low-dimensional topology. On the other hand, Geometric Representation Theory is a subject in mathematics that studies a linear model of various types of symmetries using powerful techniques of algebraic geometry. In recent years, there has been much progress relating the two subjects, enriching the subject of Geometric Representation Theory. In this colloquium style talk, we will review some recent advancements on the topic. No prior knowledge of either Quantum Field Theory or Geometric Representation Theory will be assumed.

The task of manipulating randomness has been a subject of intense investigation in the theory of computer science. The classical definition of this task consider a single processor massaging random samples from an unknown source and trying to convert it into a sequence of uniform independent bits.

In this talk I will talk about a less studied setting where randomness is distributed among different players who would like to convert this randomness to others forms with relatively little communication. For instance players may be given access to a source of biased correlated bits, and their goal may be to get a common random bit out of this source. Even in the setting where the source is known this can lead to some interesting questions that have been explored since the 70s with striking constructions and some surprisingly hard questions. After giving some background, I will describe a recent work which explores the task of extracting common randomness from correlated sources with bounds on the number of rounds of interaction.

Based on joint works with Mitali Bafna (Harvard), Badih Ghazi (Google) and Noah Golowich (Harvard).

In the last 35 years, geometric flows have proven to be a powerful tool in geometry and topology. The Mean Curvature Flow is, in many ways, the most natural flow for hypersurfaces in Euclidean space. In this talk, which will assume no prior knowledge, I will present recent progress in classifying ancient solutions to the mean curvature flow (including joints work with Kyeongsu Choi, Robert Haslhofer and Brian White). I will also explain how this classification assists in answering fundamental questions regarding the singularity formation of the flow, and describe what are the remaining challenges in converting the mean curvature flow into the powerful tool we hope it can become.

May 1-3, 2020 Harvard University Science Center, Hall B Confirmed speakers: Toby Colding, MIT Tristan Collins, MIT Hélène Esnault, Freie Universität Berlin Kenji Fukaya, Stony... Read
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May 22-24, 2020 Harvard University, Science Center Conference Poster Organizers: Laura DeMarco (Northwestern University) Sarah Koch (University of Michigan) Ronen Mukamel (Rice University) Kevin Pilgrim... Read
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November 22 - 23, 2019 Harvard University Science Center, Hall C Organized By: David Jerison, Paul Seidel, Nike Sun (MIT) Denis Auroux, Mark Kisin, Lauren... Read
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Harvard Mathematics Department graduate student Alexander Smith, who is expected to receive his Ph.D. in 2020, was awarded the 2019 inaugural David Goss Prize in Number Theory at the JNT Biennial conference in Cetraro, Italy. The newly established David Goss Prize (10K USD) will be awarded every two years to... Read more