Difference between revisions of "Colloquia/Spring2020"
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Abstract: Products of large random matrices appear in many modern applications such as high dimensional statistics (MANOVA estimators), machine learning (Jacobians of neural networks), and population ecology (transition matrices of dynamical systems). Inspired by these situations, this talk concerns global limits and fluctuations of singular values of products of independent random matrices as both the size N and number M of matrices grow. As N grows, I will show for a variety of ensembles that fluctuations of the Lyapunov exponents converge to explicit Gaussian fields which transition from log-correlated for fixed M to having a white noise component for M growing with N. I will sketch our method, which uses multivariate generalizations of the Laplace transform based on the multivariate Bessel function from representation theory. | Abstract: Products of large random matrices appear in many modern applications such as high dimensional statistics (MANOVA estimators), machine learning (Jacobians of neural networks), and population ecology (transition matrices of dynamical systems). Inspired by these situations, this talk concerns global limits and fluctuations of singular values of products of independent random matrices as both the size N and number M of matrices grow. As N grows, I will show for a variety of ensembles that fluctuations of the Lyapunov exponents converge to explicit Gaussian fields which transition from log-correlated for fixed M to having a white noise component for M growing with N. I will sketch our method, which uses multivariate generalizations of the Laplace transform based on the multivariate Bessel function from representation theory. | ||
+ | |||
+ | === Zhenfu Wang (University of Pennsylvania) === | ||
+ | |||
+ | Title: Quantitative Methods for the Mean Field Limit Problem | ||
+ | |||
+ | Abstract: We study the mean field limit of large systems of interacting particles. Classical mean field limit results require that the interaction kernels be essentially Lipschitz. To handle more singular interaction kernels is a longstanding and challenging question but which now has some successes. Joint with P.-E. Jabin, we use the relative entropy between the joint law of all particles and the tensorized law at the limit to quantify the convergence from the particle systems towards the macroscopic PDEs. This method requires to prove large deviations estimates for non-continuous potentials modified by the limiting law. But it leads to explicit convergence rates for all marginals. This in particular can be applied to the Biot-Savart law for 2D Navier-Stokes. To treat more general and singular kernels, joint with D. Bresch and P.-E. Jabin, we introduce the modulated free energy, combination of the relative entropy that we had previously developed and of the modulated energy introduced by S. Serfaty. This modulated free energy may be understood as introducing appropriate weights in the relative entropy to cancel the most singular terms involving the divergence of the kernels. Our modulated free energy allows to treat gradient flows with singular potentials which combine large smooth part, small attractive singular part and large repulsive singular part. As an example, a full rigorous derivation (with quantitative estimates) of some chemotaxis models, such as the Patlak-Keller-Segel system in the subcritical regimes, is obtained. | ||
== Future Colloquia == | == Future Colloquia == |
Revision as of 13:37, 10 February 2020
Contents
- 1 Mathematics Colloquium
- 1.1 Fall 2019
- 1.2 Spring 2020
- 1.3 Abstracts
- 1.3.1 Will Sawin (Columbia)
- 1.3.2 Yan Soibelman (Kansas State)
- 1.3.3 Alicia Dickenstein (Buenos Aires)
- 1.3.4 Jianfeng Lu (Duke)
- 1.3.5 Eugenia Cheng (School of the Art Institute of Chicago)
- 1.3.6 Omer Mermelstein (Madison)
- 1.3.7 Shamgar Gurevich (Madison)
- 1.3.8 Jose Rodriguez (UW-Madison)
- 1.3.9 Ananth Shankar (MIT)
- 1.3.10 Franca Hoffman (Caltech)
- 1.3.11 Jeffrey Danciger (UT Austin)
- 1.3.12 Tatyana Shcherbina (Princeton)
- 1.3.13 Tingran Gao (University of Chicago)
- 1.3.14 Andrew Zimmer (LSU)
- 1.3.15 Charlotte Chan (MIT)
- 1.3.16 Hui Yu (Columbia)
- 1.3.17 Alex Waldron (Michigan)
- 1.3.18 Nick Higham (Manchester)
- 1.3.19 Chenxi Wu (Rutgers)
- 1.3.20 Ruobing Zhang (Stony Brook)
- 1.3.21 Thomas Lam (Michigan)
- 1.3.22 Peter Cholak (Notre Dame)
- 1.3.23 Saulo Orizaga (Duke)
- 1.3.24 Caglar Uyanik (Yale)
- 1.3.25 Andy Zucker (Lyon)
- 1.3.26 Lillian Pierce (Duke)
- 1.3.27 Joe Kileel (Princeton)
- 1.3.28 Cynthia Vinzant (NCSU)
- 1.3.29 Jinzi Mac Huang (UCSD)
- 1.3.30 William Chan (University of North Texas)
- 1.3.31 Yi Sun (Columbia)
- 1.3.32 Zhenfu Wang (University of Pennsylvania)
- 1.4 Future Colloquia
- 1.5 Past Colloquia
Mathematics Colloquium
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, unless otherwise indicated.
Fall 2019
date | speaker | title | host(s) | |
---|---|---|---|---|
Sept 6 Room 911 | Will Sawin (Columbia) | On Chowla's Conjecture over F_q[T] | Marshall | |
Sept 13 | Yan Soibelman (Kansas State) | Riemann-Hilbert correspondence and Fukaya categories | Caldararu | |
Sept 16 Monday Room 911 | Alicia Dickenstein (Buenos Aires) | Algebra and geometry in the study of enzymatic cascades | Craciun | |
Sept 20 | Jianfeng Lu (Duke) | How to "localize" the computation? | Qin | |
Sept 26 Thursday 3-4 pm Room 911 | Eugenia Cheng (School of the Art Institute of Chicago) | Character vs gender in mathematics and beyond | Marshall / Friends of UW Madison Libraries | |
Sept 27 | ||||
Oct 4 | ||||
Oct 11 | Omer Mermelstein (Madison) | Generic flat pregeometries | Andrews | |
Oct 18 | Shamgar Gurevich (Madison) | Harmonic Analysis on GL(n) over Finite Fields | Marshall | |
Oct 25 | ||||
Nov 1 | Elchanan Mossel (MIT) | Distinguished Lecture | Roch | |
Nov 8 | Jose Rodriguez (UW-Madison) | Nearest Point Problems and Euclidean Distance Degrees | Erman | |
Nov 13 Wednesday 4-5pm | Ananth Shankar (MIT) | Exceptional splitting of abelian surfaces | ||
Nov 20 Wednesday 4-5pm | Franca Hoffman (Caltech) | Gradient Flows: From PDE to Data Analysis | Smith | |
Nov 22 | Jeffrey Danciger (UT Austin) | "Affine geometry and the Auslander Conjecture" | Kent | |
Nov 25 Monday 4-5 pm Room 911 | Tatyana Shcherbina (Princeton) | "Random matrix theory and supersymmetry techniques" | Roch | |
Nov 29 | Thanksgiving | |||
Dec 2 Monday 4-5pm | Tingran Gao (University of Chicago) | "Manifold Learning on Fibre Bundles" | Smith | |
Dec 4 Wednesday 4-5 pm Room 911 | Andrew Zimmer (LSU) | "Intrinsic and extrinsic geometries in several complex variables" | Gong | |
Dec 6 | Charlotte Chan (MIT) | "Flag varieties and representations of p-adic groups" | Erman | |
Dec 9 Monday 4-5 pm | Hui Yu (Columbia) | Singular sets in obstacle problems | Tran | |
Dec 11 Wednesday 2:30-3:30pm Room 911 | Alex Waldron (Michigan) | Gauge theory and geometric flows | Paul | |
Dec 11 Wednesday 4-5pm | Nick Higham (Manchester) | LAA lecture: Challenges in Multivalued Matrix Functions | Brualdi | |
Dec 13 | Chenxi Wu (Rutgers) | Kazhdan's theorem on metric graphs | Ellenberg | |
Dec 18 Wednesday 4-5pm | Ruobing Zhang (Stony Brook) | Geometry and analysis of degenerating Calabi-Yau manifolds | Paul |
Spring 2020
date | speaker | title | host(s) | |
---|---|---|---|---|
Jan 10 | Thomas Lam (Michigan) | Positive geometries and string theory amplitudes | Erman | |
Jan 21 Tuesday 4-5 pm in B139 | Peter Cholak (Notre Dame) | What can we compute from solutions to combinatorial problems? | Lempp | |
Jan 24 | Saulo Orizaga (Duke) | Introduction to phase field models and their efficient numerical implementation | ||
Jan 27 Monday 4-5 pm in 911 | Caglar Uyanik (Yale) | Hausdorff dimension and gap distribution in billiards | Ellenberg | |
Jan 29 Wednesday 4-5 pm | Andy Zucker (Lyon) | Topological dynamics of countable groups and structures | Soskova/Lempp | |
Jan 31 | Lillian Pierce (Duke) | On Bourgain’s counterexample for the Schrödinger maximal function | Marshall/Seeger | |
Feb 7 | Joe Kileel (Princeton) | Inverse Problems, Imaging and Tensor Decomposition | Roch | |
Feb 10 | Cynthia Vinzant (NCSU) | Matroids, log-concavity, and expanders | Roch/Erman | |
Feb 12 Wednesday 4-5 pm in VV 911 | Jinzi Mac Huang (UCSD) | Mass transfer through fluid-structure interactions | Spagnolie | |
Feb 14 | William Chan (University of North Texas) | Definable infinitary combinatorics under determinacy | Soskova/Lempp | |
Feb 17 | Yi Sun (Columbia) | Fluctuations for products of random matrices | Roch | |
Feb 19 | Zhenfu Wang (University of Pennsylvania) | Quantitative Methods for the Mean Field Limit Problem | Tran | |
Feb 21 | Shai Evra (IAS) | Gurevich | ||
Feb 28 | Brett Wick (Washington University, St. Louis) | Seeger | ||
March 6 | Jessica Fintzen (Michigan) | Marshall | ||
March 13 | ||||
March 20 | Spring break | |||
March 27 | Max Lieblich (Univ. of Washington, Seattle) | Boggess, Sankar | ||
April 3 | Caroline Turnage-Butterbaugh (Carleton College) | Marshall | ||
April 10 | Sarah Koch (Michigan) | Bruce (WIMAW) | ||
April 17 | JM Landsberg (TAMU) | TBA | Gurevich | |
April 23 | Martin Hairer (Imperial College London) | Wolfgang Wasow Lecture | Hao Shen | |
April 24 | Natasa Sesum (Rutgers University) | Angenent | ||
May 1 | Robert Lazarsfeld (Stony Brook) | Distinguished lecture | Erman |
Abstracts
Will Sawin (Columbia)
Title: On Chowla's Conjecture over F_q[T]
Abstract: The Mobius function in number theory is a sequences of 1s, -1s, and 0s, which is simple to define and closely related to the prime numbers. Its behavior seems highly random. Chowla's conjecture is one precise formalization of this randomness, and has seen recent work by Matomaki, Radziwill, Tao, and Teravainen making progress on it. In joint work with Mark Shusterman, we modify this conjecture by replacing the natural numbers parameterizing this sequence with polynomials over a finite field. Under mild conditions on the finite field, we are able to prove a strong form of this conjecture. The proof is based on taking a geometric perspective on the problem, and succeeds because we are able to simplify the geometry using a trick based on the strange properties of polynomial derivatives over finite fields.
Yan Soibelman (Kansas State)
Title: Riemann-Hilbert correspondence and Fukaya categories
Abstract: In this talk I am going to discuss the role of Fukaya categories in the Riemann-Hilbert correspondence for differential, q-difference and elliptic difference equations in dimension one. This approach not only gives a unified answer for several versions of the Riemann-Hilbert correspondence but also leads to a natural formulation of the non-abelian Hodge theory in dimension one. It also explains why periodic monopoles should appear as harmonic objects in this generalized non-abelian Hodge theory. All that is a part of the bigger project ``Holomorphic Floer theory", joint with Maxim Kontsevich.
Alicia Dickenstein (Buenos Aires)
Title: Algebra and geometry in the study of enzymatic cascades
Abstract: In recent years, techniques from computational and real algebraic geometry have been successfully used to address mathematical challenges in systems biology. The algebraic theory of chemical reaction systems aims to understand their dynamic behavior by taking advantage of the inherent algebraic structure in the kinetic equations, and does not need the determination of the parameters a priori, which can be theoretically or practically impossible. I will give a gentle introduction to general results based on the network structure. In particular, I will describe a general framework for biological systems, called MESSI systems, that describe Modifications of type Enzyme-Substrate or Swap with Intermediates, and include many networks that model post-translational modifications of proteins inside the cell. I will also outline recent methods to address the important question of multistationarity, in particular in the study of enzymatic cascades, and will point out some of the mathematical challenges that arise from this application.
Jianfeng Lu (Duke)
Title: How to ``localize" the computation?
It is often desirable to restrict the numerical computation to a local region to achieve best balance between accuracy and affordability in scientific computing. It is important to avoid artifacts and guarantee predictable modelling while artificial boundary conditions have to be introduced to restrict the computation. In this talk, we will discuss some recent understanding on how to achieve such local computation in the context of topological edge states and elliptic random media.
Eugenia Cheng (School of the Art Institute of Chicago)
Title: Character vs gender in mathematics and beyond
Abstract: This presentation will be based on my experience of being a female mathematician, and teaching mathematics at all levels from elementary school to grad school. The question of why women are under-represented in mathematics is complex and there are no simple answers, only many many contributing factors. I will focus on character traits, and argue that if we focus on this rather than gender we can have a more productive and less divisive conversation. To try and focus on characters rather than genders I will introduce gender-neutral character adjectives "ingressive" and "congressive" to replace masculine and feminine. I will share my experience of teaching congressive abstract mathematics to art students, in a congressive way, and the possible effects this could have for everyone in mathematics, not just women.
Omer Mermelstein (Madison)
Title: Generic flat pregeometries
Abstract: In model theory, the tamest of structures are the strongly minimal ones -- those in which every equation in a single variable has either finitely many or cofinitely many solution. Algebraically closed fields and vector spaces are the canonical examples. Zilber’s conjecture, later refuted by Hrushovski, states that the source of geometric complexity in a strongly minimal structure must be algebraic. The property of "flatness" (strict gammoid) of a geometry (matroid) is that which guarantees Hrushovski's construction is devoid of any associative structure. The majority of the talk will explain what flatness is, how it should be thought of, and how closely it relates to hypergraphs and Hrushovski's construction method. Model theory makes an appearance only in the second part, where I will share results pertaining to the specific family of geometries arising from Hrushovski's methods.
Shamgar Gurevich (Madison)
Title: Harmonic Analysis on GL(n) over Finite Fields.
Abstract: There are many formulas that express interesting properties of a finite group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the character ratio:
trace(ρ(g)) / dim(ρ),
for an irreducible representation ρ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of the mentioned type for analyzing certain random walks on G.
Recently, we discovered that for classical groups G over finite fields there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant rank.
This talk will discuss the notion of rank for the group GLn over finite fields, demonstrate how it controls the character ratio, and explain how one can apply the results to verify mixing time and rate for certain random walks.
This is joint work with Roger Howe (Yale and Texas AM). The numerics for this work was carried by Steve Goldstein (Madison)
Jose Rodriguez (UW-Madison)
Abstract: Determining the closest point to a model (subset of Euclidean space) is an important problem in many applications in science, engineering, and statistics. One way to solve this problem is by minimizing the squared Euclidean distance function using a gradient descent approach. However, when there are multiple local minima, there is no guarantee of convergence to the true global minimizer. An alternative method is to determine the critical points of an objective function on the model. In algebraic statistics, the models of interest are algebraic sets, i.e., solution sets to a system of multivariate polynomial equations. In this situation, the number of critical points of the squared Euclidean distance function on the model’s Zariski closure is a topological invariant called the Euclidean distance degree (ED degree). In this talk, I will present some models from computer vision and statistics that may be described as algebraic sets. Moreover, I will describe a topological method for determining a Euclidean distance degree and a numerical algebraic geometry approach for determining critical points of the squared Euclidean distance function.
Ananth Shankar (MIT)
Abstract: An abelian surface 'splits' if it admits a non-trivial map to some elliptic curve. It is well known that the set of abelian surfaces that split are sparse in the set of all abelian surfaces. Nevertheless, we prove that there are infinitely many split abelian surfaces in arithmetic one-parameter families of generically non-split abelian surfaces. I will describe this work, and if time permits, mention generalizations of this result to the setting of K3 surfaces, as well as applications to the dynamics of hecke orbits. This is joint work with Tang, Maulik-Tang, and Shankar-Tang-Tayou.
Franca Hoffman (Caltech)
Title: Gradient Flows: From PDE to Data Analysis.
Abstract: Certain diffusive PDEs can be viewed as infinite-dimensional gradient flows. This fact has led to the development of new tools in various areas of mathematics ranging from PDE theory to data science. In this talk, we focus on two different directions: model-driven approaches and data-driven approaches. In the first part of the talk we use gradient flows for analyzing non-linear and non-local aggregation-diffusion equations when the corresponding energy functionals are not necessarily convex. Moreover, the gradient flow structure enables us to make connections to well-known functional inequalities, revealing possible links between the optimizers of these inequalities and the equilibria of certain aggregation-diffusion PDEs. In the second part, we use and develop gradient flow theory to design novel tools for data analysis. We draw a connection between gradient flows and Ensemble Kalman methods for parameter estimation. We introduce the Ensemble Kalman Sampler - a derivative-free methodology for model calibration and uncertainty quantification in expensive black-box models. The interacting particle dynamics underlying our algorithm can be approximated by a novel gradient flow structure in a modified Wasserstein metric which reflects particle correlations. The geometry of this modified Wasserstein metric is of independent theoretical interest.
Jeffrey Danciger (UT Austin)
Title: Affine geometry and the Auslander Conjecture
Abstract: The Auslander Conjecture is an analogue of Bieberbach’s theory of Euclidean crystallographic groups in the setting of affine geometry. It predicts that a complete affine manifold (a manifold equipped with a complete torsion-free flat affine connection) which is compact must have virtually solvable fundamental group. The conjecture is known up to dimension six, but is known to fail if the compactness assumption is removed, even in low dimensions. We discuss some history of this conjecture, give some basic examples, and then survey some recent advances in the study of non-compact complete affine manifolds with non-solvable fundamental group. Tools from the deformation theory of pseudo-Riemannian hyperbolic manifolds and also from higher Teichmüller theory will enter the picture.
Tatyana Shcherbina (Princeton)
Title: Random matrix theory and supersymmetry techniques
Abstract: Starting from the works of Erdos, Yau, Schlein with coauthors, the significant progress in understanding the universal behavior of many random graph and random matrix models were achieved. However for the random matrices with a special structure our understanding is still very limited. In this talk I am going to overview applications of another approach to the study of the local eigenvalues statistics in random matrix theory based on so-called supersymmetry techniques (SUSY). SUSY approach is based on the representation of the determinant as an integral over the Grassmann (anticommuting) variables. Combining this representation with the representation of an inverse determinant as an integral over the Gaussian complex field, SUSY allows to obtain an integral representation for the main spectral characteristics of random matrices such as limiting density, correlation functions, the resolvent's elements, etc. This method is widely (and successfully) used in the physics literature and is potentially very powerful but the rigorous control of the integral representations, which can be obtained by this method, is quite difficult, and it requires powerful analytic and statistical mechanics tools. In this talk we will discuss some recent progress in application of SUSY to the analysis of local spectral characteristics of the prominent ensemble of random band matrices, i.e. random matrices whose entries become negligible if their distance from the main diagonal exceeds a certain parameter called the band width.
Tingran Gao (University of Chicago)
Title: Manifold Learning on Fibre Bundles
Abstract: Spectral geometry has played an important role in modern geometric data analysis, where the technique is widely known as Laplacian eigenmaps or diffusion maps. In this talk, we present a geometric framework that studies graph representations of complex datasets, where each edge of the graph is equipped with a non-scalar transformation or correspondence. This new framework models such a dataset as a fibre bundle with a connection, and interprets the collection of pairwise functional relations as defining a horizontal diffusion process on the bundle driven by its projection on the base. The eigenstates of this horizontal diffusion process encode the “consistency” among objects in the dataset, and provide a lens through which the geometry of the dataset can be revealed. We demonstrate an application of this geometric framework on evolutionary anthropology.
Andrew Zimmer (LSU)
Title: Intrinsic and extrinsic geometries in several complex variables
Abstract: A bounded domain in complex Euclidean space, despite being one of the simplest types of manifolds, has a number of interesting geometric structures. When the domain is pseudoconvex, it has a natural intrinsic geometry: the complete Kaehler-Einstein metric constructed by Cheng-Yau and Mok-Yau. When the domain is smoothly bounded, there is also a natural extrinsic structure: the CR-geometry of the boundary. In this talk, I will describe connections between these intrinsic and extrinsic geometries. Then, I will discuss how these connections can lead to new analytic results.
Charlotte Chan (MIT)
Title: Flag varieties and representations of p-adic groups
Abstract: In the 1950s, Borel, Weil, and Bott showed that the irreducible representations of a complex reductive group can be realized in the cohomology of line bundles on flag varieties. In the 1970s, Deligne and Lusztig constructed a family of subvarieties of flag varieties whose cohomology realizes the irreducible representations of reductive groups over finite fields. I will survey these stories, explain recent progress towards finding geometric constructions of representations of p-adic groups, and discuss interactions with the Langlands program.
Hui Yu (Columbia)
Title: Singular sets in obstacle problems
Abstract: One of the most important free boundary problems is the obstacle problem. The regularity of its free boundary has been studied for over half a century. In this talk, we review some classical results as well as exciting new developments. In particular, we discuss the recent resolution of the regularity of the singular set for the fully nonlinear obstacle problem. This talk is based on a joint work with Ovidiu Savin at Columbia University.
Alex Waldron (Michigan)
Title: Gauge theory and geometric flows
Abstract: I will give a brief introduction to two major areas of research in differential geometry: gauge theory and geometric flows. I'll then introduce a geometric flow (Yang-Mills flow) arising from a variational problem with origins in physics, which has been studied by geometric analysts since the early 1980s. I'll conclude by discussing my own work on the behavior of Yang-Mills flow in the critical dimension (n = 4).
Nick Higham (Manchester)
Title: Challenges in Multivalued Matrix Functions
Abstract: In this lecture I will discuss multivalued matrix functions that arise in solving various kinds of matrix equations. The matrix logarithm is the prototypical example, and my first interaction with Hans Schneider was about this function. Another example is the Lambert W function of a matrix, which is much less well known but has been attracting recent interest. A theme of the talk is the importance of choosing appropriate principal values and making sure that the correct choices of signs and branches are used, both in theory and in computation. I will give examples where incorrect results have previously been obtained.
I focus on matrix inverse trigonometric and inverse hyperbolic functions, beginning by investigating existence and characterization. Turning to the principal values, various functional identities are derived, some of which are new even in the scalar case, including a “round trip” formula that relates acos(cos A) to A and similar formulas for the other inverse functions. Key tools used in the derivations are the matrix unwinding function and the matrix sign function.
A new inverse scaling and squaring type algorithm employing a Schur decomposition and variable-degree Pade approximation is derived for computing acos, and it is shown how it can also be used to compute asin, acosh, and asinh.
Chenxi Wu (Rutgers)
Title: Kazhdan's theorem on metric graphs
Abstract: I will give an introduction to the concept of canonical (arakelov) metric on a metric graph, which is related to combinatorial questions like the counting of spanning trees, and generalizes the corresponding concept on Riemann surfaces. I will also present a recent result in collaboration with Farbod Shokrieh on the convergence of canonical metric under normal covers.
Ruobing Zhang (Stony Brook)
Title: Geometry and analysis of degenerating Calabi-Yau manifolds
Abstract: This talk concerns a naturally occurring family of degenerating Calabi-Yau manifolds. A primary tool in analyzing their behavior is to combine the recently developed structure theory for Einstein manifolds and multi-scale singularity analysis for degenerating nonlinear PDEs in the collapsed setting. Based on the algebraic degeneration, we will give precise and more quantitative descriptions of singularity formation from both metric and analytic points of view.
Thomas Lam (Michigan)
Title: Positive geometries and string theory amplitudes
Abstract: Inspired by developments in quantum field theory, we recently defined the notion of a positive geometry, a class of spaces that includes convex polytopes, positive parts of projective toric varieties, and positive parts of flag varieties. I will discuss some basic features of the theory and an application to genus zero string theory amplitudes. As a special case, we obtain the Euler beta function, familiar to mathematicians, as the "stringy canonical form" of the closed interval.
This talk is based on joint work with Arkani-Hamed, Bai, and He.
Peter Cholak (Notre Dame)
Title: What can we compute from solutions to combinatorial problems?
Abstract: This will be an introductory talk to an exciting current research area in mathematical logic. Mostly we are interested in solutions to Ramsey's Theorem. Ramsey's Theorem says for colorings C of pairs of natural numbers, there is an infinite set H such that all pairs from H have the same constant color. H is called a homogeneous set for C. What can we compute from H? If you are not sure, come to the talk and find out!
Saulo Orizaga (Duke)
Title: Introduction to phase field models and their efficient numerical implementation
Abstract: In this talk we will provide an introduction to phase field models. We will focus in models related to the Cahn-Hilliard (CH) type of partial differential equation (PDE). We will discuss the challenges associated in solving such higher order parabolic problems. We will present several new numerical methods that are fast and efficient for solving CH or CH-extended type of problems. The new methods and their energy-stability properties will be discussed and tested with several computational examples commonly found in material science problems. If time allows, we will talk about more applications in which phase field models are useful and applicable.
Caglar Uyanik (Yale)
Title: Hausdorff dimension and gap distribution in billiards
Abstract: A classical “unfolding” procedure allows one to turn questions about billiard trajectories in a Euclidean polygon into questions about the geodesic flow on a surface equipped with a certain geometric structure. Surprisingly, the flow on the surface is in turn related to the geodesic flow on the classical moduli spaces of Riemann surfaces. Building on recent breakthrough results of Eskin-Mirzakhani-Mohammadi, we prove a large deviations result for Birkhoff averages as well as generalize a classical theorem of Masur on geodesics in the moduli spaces of translation surfaces.
Andy Zucker (Lyon)
Title: Topological dynamics of countable groups and structures
Abstract: We give an introduction to the abstract topological dynamics of topological groups, i.e. the study of the continuous actions of a topological group on a compact space. We are particularly interested in the minimal actions, those for which every orbit is dense. The study of minimal actions is aided by a classical theorem of Ellis, who proved that for any topological group G, there exists a universal minimal flow (UMF), a minimal G-action which factors onto every other minimal G-action. Here, we will focus on two classes of groups: a countable discrete group and the automorphism group of a countable first-order structure. In the case of a countable discrete group, Baire category methods can be used to show that the collection of minimal flows is quite rich and that the UMF is rather complicated. For an automorphism group G of a countable structure, combinatorial methods can be used to show that sometimes, the UMF is trivial, or equivalently that every continuous action of G on a compact space admits a global fixed point.
Lillian Pierce (Duke)
Title: On Bourgain’s counterexample for the Schrödinger maximal function
Abstract: In 1980, Carleson asked a question in harmonic analysis: to which Sobolev space $H^s$ must an initial data function belong, for a pointwise a.e. convergence result to hold for the solution to the associated linear Schrödinger equation? Over the next decades, many people developed counterexamples to push the (necessary) range of s up, and positive results to push the (sufficient) range of s down. Now, these ranges are finally meeting: Bourgain’s 2016 counterexample showed s < n/(2(n+1)) fails, and Du and Zhang’s 2019 paper shows that s>n/(2(n+1)) suffices. In this talk, we will give an overview of how to rigorously derive Bourgain’s 2016 counterexample, based on simple facts from number theory. We will show how to build Bourgain’s counterexample starting from “zero knowledge," and how to gradually optimize the set-up to arrive at the final counterexample. The talk will be broadly accessible, particularly if we live up to the claim of starting from “zero knowledge.”
Joe Kileel (Princeton)
Title: Inverse Problems, Imaging and Tensor Decomposition
Abstract: Perspectives from computational algebra and optimization are brought to bear on a scientific application and a data science application. In the first part of the talk, I will discuss cryo-electron microscopy (cryo-EM), an imaging technique to determine the 3-D shape of macromolecules from many noisy 2-D projections, recognized by the 2017 Chemistry Nobel Prize. Mathematically, cryo-EM presents a particularly rich inverse problem, with unknown orientations, extreme noise, big data and conformational heterogeneity. In particular, this motivates a general framework for statistical estimation under compact group actions, connecting information theory and group invariant theory. In the second part of the talk, I will discuss tensor rank decomposition, a higher-order variant of PCA broadly applicable in data science. A fast algorithm is introduced and analyzed, combining ideas of Sylvester and the power method.
Cynthia Vinzant (NCSU)
Title: Matroids, log-concavity, and expanders
Abstract: Matroids are combinatorial objects that model various types of independence. They appear several fields mathematics, including graph theory, combinatorial optimization, and algebraic geometry. In this talk, I will introduce the theory of matroids along with the closely related class of polynomials called strongly log-concave polynomials. Strong log-concavity is a functional property of a real multivariate polynomial that translates to useful conditions on its coefficients. Discrete probability distributions defined by these coefficients inherit several of these nice properties. I will discuss the beautiful real and combinatorial geometry underlying these polynomials and describe applications to random walks on the faces of simplicial complexes. Consequences include proofs of Mason's conjecture that the sequence of numbers of independent sets of a matroid is ultra log-concave and the Mihail-Vazirani conjecture that the basis exchange graph of a matroid has expansion at least one. This is based on joint work with Nima Anari, Kuikui Liu, and Shayan Oveis Gharan.
Jinzi Mac Huang (UCSD)
Title: Mass transfer through fluid-structure interactions
Abstract: The advancement of mathematics is closely associated with new discoveries from physical experiments. On one hand, mathematical tools like numerical simulation can help explain observations from experiments. On the other hand, experimental discoveries of physical phenomena, such as Brownian motion, can inspire the development of new mathematical approaches. In this talk, we focus on the interplay between applied math and experiments involving fluid-structure interactions -- a fascinating topic with both physical relevance and mathematical complexity. One such problem, inspired by geophysical fluid dynamics, is the experimental and numerical study of the dissolution of solid bodies in a fluid flow. The results of this study allow us to sketch mathematical answers to some long standing questions like the formation of stone forests in China and Madagascar, and how many licks it takes to get to the center of a Tootsie Pop. We will also talk about experimental math problems at the micro-scale, focusing on the mass transport process of diffusiophoresis, where colloidal particles are advected by a concentration gradient of salt solution. Exploiting this phenomenon, we see that colloids are able to navigate a micro-maze that has a salt concentration gradient across the exit and entry points. We further demonstrate that their ability to solve the maze is closely associated with the properties of a harmonic function – the salt concentration.
William Chan (University of North Texas)
Title: Definable infinitary combinatorics under determinacy
Abstract: The axiom of determinacy, AD, states that in any infinite two player integer game of a certain form, one of the two players must have a winning strategy. It is incompatible with the ZFC set theory axioms with choice; however, it is a succinct extension of ZF which implies many subsets of the real line possess familiar regularity properties and eliminates many pathological sets. For instance, AD implies all sets of reals are Lebesgue measurable and every function from the reals to the reals is continuous on a comeager set. Determinacy also implies that the first uncountable cardinal has the strong partition property which can be used to define the partition measures. This talk will give an overview of the axiom of determinacy and will discuss recent results on the infinitary combinatorics surrounding the first uncountable cardinal and its partition measures. I will discuss the almost everywhere continuity phenomenon for functions outputting countable ordinals and the almost-everywhere uniformization results for closed and unbounded subsets of the first uncountable cardinal. These will be used to describe the rich structure of the cardinals below the powerset of the first and second uncountable cardinals under determinacy assumptions and to investigate the ultrapowers by these partition measures.
Yi Sun (Columbia)
Title: Fluctuations for products of random matrices
Abstract: Products of large random matrices appear in many modern applications such as high dimensional statistics (MANOVA estimators), machine learning (Jacobians of neural networks), and population ecology (transition matrices of dynamical systems). Inspired by these situations, this talk concerns global limits and fluctuations of singular values of products of independent random matrices as both the size N and number M of matrices grow. As N grows, I will show for a variety of ensembles that fluctuations of the Lyapunov exponents converge to explicit Gaussian fields which transition from log-correlated for fixed M to having a white noise component for M growing with N. I will sketch our method, which uses multivariate generalizations of the Laplace transform based on the multivariate Bessel function from representation theory.
Zhenfu Wang (University of Pennsylvania)
Title: Quantitative Methods for the Mean Field Limit Problem
Abstract: We study the mean field limit of large systems of interacting particles. Classical mean field limit results require that the interaction kernels be essentially Lipschitz. To handle more singular interaction kernels is a longstanding and challenging question but which now has some successes. Joint with P.-E. Jabin, we use the relative entropy between the joint law of all particles and the tensorized law at the limit to quantify the convergence from the particle systems towards the macroscopic PDEs. This method requires to prove large deviations estimates for non-continuous potentials modified by the limiting law. But it leads to explicit convergence rates for all marginals. This in particular can be applied to the Biot-Savart law for 2D Navier-Stokes. To treat more general and singular kernels, joint with D. Bresch and P.-E. Jabin, we introduce the modulated free energy, combination of the relative entropy that we had previously developed and of the modulated energy introduced by S. Serfaty. This modulated free energy may be understood as introducing appropriate weights in the relative entropy to cancel the most singular terms involving the divergence of the kernels. Our modulated free energy allows to treat gradient flows with singular potentials which combine large smooth part, small attractive singular part and large repulsive singular part. As an example, a full rigorous derivation (with quantitative estimates) of some chemotaxis models, such as the Patlak-Keller-Segel system in the subcritical regimes, is obtained.