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__NOTOC__
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= Mathematics Colloquium =
 
= Mathematics Colloquium =
  
 
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.
 
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.
  
==Spring 2019==
+
 
 +
 
 +
==Spring 2020==
  
 
{| cellpadding="8"
 
{| cellpadding="8"
Line 10: Line 14:
 
!align="left" | title
 
!align="left" | title
 
!align="left" | host(s)
 
!align="left" | host(s)
 +
|
 
|-
 
|-
|Jan 25 '''Room 911'''
+
|Jan 10
| [http://www.users.miamioh.edu/randrib/ Beata Randrianantoanina] (Miami University Ohio) WIMAW
+
|Thomas Lam (Michigan)
|[[#Beata Randrianantoanina (Miami University Ohio) | Some nonlinear problems in the geometry of Banach spaces and their applications  ]]
+
|[[#Thomas Lam (Michigan) |Positive geometries and string theory amplitudes]]
| Tullia Dymarz
+
|Erman
 +
|-
 +
|Jan 21  '''Tuesday 4-5 pm in B139'''
 +
|[http://www.nd.edu/~cholak/ Peter Cholak] (Notre Dame)
 +
|[[#Peter Cholak (Notre Dame) |What can we compute from solutions to combinatorial problems?]]
 +
|Lempp
 +
|-
 +
|Jan 24
 +
|[https://math.duke.edu/people/saulo-orizaga Saulo Orizaga] (Duke)
 +
|[[#Saulo Orizaga (Duke) | Introduction to phase field models and their efficient numerical implementation ]]
 
|
 
|
 
|-
 
|-
|Jan 30 '''Wednesday'''
+
|Jan 27 '''Monday 4-5 pm in 911'''
| Talk rescheduled to Feb 15
+
|[https://math.yale.edu/people/caglar-uyanik Caglar Uyanik] (Yale)
|
+
|[[#Caglar Uyanik (Yale) | Hausdorff dimension and gap distribution in billiards ]]
 +
|Ellenberg
 +
|-
 +
|Jan 29  '''Wednesday 4-5 pm'''
 +
|[https://ajzucker.wordpress.com/ Andy Zucker] (Lyon)
 +
|[[#Andy Zucker (Lyon) |Topological dynamics of countable groups and structures]]
 +
|Soskova/Lempp
 
|-
 
|-
|Jan 31 '''Thursday'''
+
|Jan 31  
| Talk rescheduled to Feb 13
+
|[https://services.math.duke.edu/~pierce/ Lillian Pierce] (Duke)
|
+
|[[#Lillian Pierce (Duke) |On Bourgain’s counterexample for the Schrödinger maximal function]]
 +
|Marshall/Seeger
 
|-
 
|-
|Feb 1
+
|Feb 7
| Talk cancelled due to weather
+
|[https://web.math.princeton.edu/~jkileel/ Joe Kileel] (Princeton)
|
+
|[[#Joe Kileel (Princeton) |Inverse Problems, Imaging and Tensor Decomposition]]
|  
+
|Roch
|
 
 
|-
 
|-
|Feb 5 '''Tuesday, VV 911'''
+
|Feb 10
| [http://www.math.tamu.edu/~alexei.poltoratski/ Alexei Poltoratski] (Texas A&M University)
+
|[https://clvinzan.math.ncsu.edu/ Cynthia Vinzant] (NCSU)
|[[#Alexei Poltoratski (Texas A&M)| Completeness of exponentials: Beurling-Malliavin and type problems  ]]
+
|[[#Cynthia Vinzant (NCSU) |Matroids, log-concavity, and expanders]]
| Denisov
+
|Roch/Erman
|
 
 
|-
 
|-
|Feb 6 '''Wednesday, room 911'''
+
|Feb 12 '''Wednesday 4-5 pm in VV 911'''
| [https://lc-tsai.github.io/ Li-Cheng Tsai] (Columbia University)
+
|[https://www.machuang.org/ Jinzi Mac Huang] (UCSD)
|[[#Li-Cheng Tsai (Columbia University)| When particle systems meet PDEs  ]]
+
|[[#Jinzi Mac Huang (UCSD) |Mass transfer through fluid-structure interactions]]
| Anderson
+
|Spagnolie
|
 
 
|-
 
|-
|Feb 8
+
|Feb 14
| [https://sites.math.northwestern.edu/~anaber/ Aaron Naber] (Northwestern)
+
|[https://math.unt.edu/people/william-chan/ William Chan] (University of North Texas)
|[[#Aaron Naber (Northwestern) |   A structure theory for spaces with lower Ricci curvature bounds  ]]
+
|[[#William Chan (University of North Texas) |Definable infinitary combinatorics under determinacy]]
| Street
+
|Soskova/Lempp
|
 
 
|-
 
|-
|Feb 11 '''Monday'''
+
|Feb 17
| [https://www2.bc.edu/david-treumann/materials.html David Treumann] (Boston College)
+
|[https://yisun.io/ Yi Sun] (Columbia)
|[[#David Treumann (Boston College) |   Twisting things in topology and symplectic topology by pth powers  ]]
+
|[[#Yi Sun (Columbia) |Fluctuations for products of random matrices]]
| Caldararu
+
|Roch
|
 
 
|-
 
|-
| Feb 13 '''Wednesday'''
+
|Feb 19
| [http://www.math.tamu.edu/~dbaskin/ Dean Baskin] (Texas A&M)
+
|[https://www.math.upenn.edu/~zwang423// Zhenfu Wang] (University of Pennsylvania)
|[[#Dean Baskin (Texas A&M) | Radiation fields for wave equations  ]]
+
|[[#Zhenfu Wang (University of Pennsylvania) |Quantitative Methods for the Mean Field Limit Problem]]
| Street
+
|Tran
 
 
 
|-
 
|-
| Feb 15
+
|Feb 21
| [https://services.math.duke.edu/~pierce/ Lillian Pierce] (Duke University)
+
|Shai Evra (IAS)
| [[#Lillian Pierce (Duke University) | Short character sums  ]]
+
|[[#Shai Evra (IAS) |Golden Gates in PU(n) and the Density Hypothesis]]
| Boston and Street
+
|Gurevich
 
|
 
|
 
|-
 
|-
|Feb 22
+
|Feb 28
| [https://people.math.osu.edu/cueto.5/ Angelica Cueto] (Ohio State)
+
|Brett Wick (Washington University, St. Louis)
|[[# Angelica Cueto| Lines on cubic surfaces in the tropics  ]]
+
|[[#Brett Wick (WUSTL) |The Corona Theorem]]
| Erman and Corey
+
|Seeger
|
 
 
|-
 
|-
|March 4
+
|March 6 '''in 911'''
| [http://www-users.math.umn.edu/~sverak/ Vladimir Sverak] (Minnesota) Wasow lecture
+
| Jessica Fintzen (Michigan)
|[[# TBA| TBA ]]
+
|[[#Jessica Fintzen (Michigan) | Representations of p-adic groups]]
| Kim
+
|Marshall
|
 
 
|-
 
|-
|March 8
+
|March 13 '''CANCELLED'''
| [https://orion.math.iastate.edu/jmccullo/index.html Jason McCullough] (Iowa State)
+
| [https://plantpath.wisc.edu/claudia-solis-lemus// Claudia Solis Lemus] (UW-Madison, Plant Pathology)
|[[# TBA| TBA  ]]
+
|[[#Claudia Solis Lemus | New challenges in phylogenetic inference]]
| Erman
+
|Anderson
|
 
 
|-
 
|-
|March 15
+
|March 20
| Maksym Radziwill (Caltech)
+
|Spring break
|[[# TBA|  TBA  ]]
 
| Marshall
 
 
|
 
|
 
|-
 
|-
|March 29
+
|March 27 '''CANCELLED'''
| Jennifer Park (OSU)
+
|[https://max.lieblich.us/ Max Lieblich] (Univ. of Washington, Seattle)
|[[# TBA|  TBA  ]]
 
| Marshall
 
 
|
 
|
 +
|Boggess, Sankar
 
|-
 
|-
|April 5
+
|April 3 '''CANCELLED'''
| Ju-Lee Kim (MIT)
+
|Caroline Turnage-Butterbaugh (Carleton College)
|[[# TBA|  TBA  ]]
 
| Gurevich
 
 
|
 
|
 +
|Marshall
 
|-
 
|-
|April 12
+
|April 10
| Eviatar Procaccia (TAMU)
+
| No colloquium
|[[# TBA|  TBA  ]]
 
| Gurevich
 
 
|
 
|
 +
|
 
|-
 
|-
|April 19
+
|April 17
| [http://www.math.rice.edu/~jkn3/ Jo Nelson] (Rice University)
+
|JM Landsberg (TAMU)
|[[# TBA|  TBA  ]]
+
|TBA
| Jean-Luc
+
|Gurevich
|
 
 
|-
 
|-
|April 22
+
|April 23
| [https://justinh.su Justin Hsu] (Madison)
+
|Martin Hairer (Imperial College London)
|[[# TBA|  TBA  ]]
+
|Wolfgang Wasow Lecture
| Steffen
+
|Hao Shen
|
 
 
|-
 
|-
|April 26
+
|April 24
| [https://www.brown.edu/academics/applied-mathematics/faculty/kavita-ramanan/home Kavita Ramanan] (Brown University)
+
|Natasa Sesum (Rutgers University)
|[[# TBA|  TBA  ]]
 
| WIMAW
 
 
|
 
|
 +
|Angenent
 
|-
 
|-
|May 3
+
|May 1
| Tomasz Przebinda (Oklahoma)
+
|Robert Lazarsfeld (Stony Brook)
|[[# TBA|  TBA  ]]
+
|Distinguished lecture
| Gurevich
+
|Erman
|
 
 
|}
 
|}
  
 
== Abstracts ==
 
== Abstracts ==
  
===Beata Randrianantoanina (Miami University Ohio)===
+
=== 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: Some nonlinear problems in the geometry of Banach spaces and their applications.
+
Title: Inverse Problems, Imaging and Tensor Decomposition
  
Abstract: Nonlinear problems in the geometry of Banach spaces have been studied since the inception of the field. In this talk I will outline some of the history, some of modern applications, and some open directions of research. The talk will be accessible to graduate students of any field of mathematics.
+
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.
  
===Lillian Pierce (Duke University)===
+
=== Cynthia Vinzant (NCSU) ===
  
Title: Short character sums
+
Title: Matroids, log-concavity, and expanders
  
Abstract: A surprisingly diverse array of problems in analytic number theory have at their heart a problem of bounding (from above) an exponential sum, or its multiplicative cousin, a so-called character sum. For example, both understanding the Riemann zeta function or Dirichlet L-functions inside the critical strip, and also counting solutions to Diophantine equations via the circle method or power sieve methods, involve bounding such sums. In general, the sums of interest fall into one of two main regimes: complete sums or incomplete sums, with this latter regime including in particular “short sums.” Short sums are particularly useful, and particularly resistant to almost all known methods. In this talk, we will see what makes a sum “short,” sketch why it would be incredibly powerful to understand short sums, and discuss a curious proof from the 1950’s which is still the best way we know to bound short sums. We will end by describing new work which extends the ideas of this curious proof to bound short sums in much more general situations.
+
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.
  
===Angelica Cueto (The Ohio State University)===
+
=== Jinzi Mac Huang (UCSD) ===
Title: Lines on cubic surfaces in the tropics
 
  
Abstract: Since the beginning of tropical geometry, a persistent challenge has been to emulate tropical versions of classical results in algebraic geometry. The well-know statement <i>any smooth surface of degree three in P^3 contains exactly 27 lines</i> is known to be false tropically. Work of Vigeland from 2007 provides examples of tropical cubic surfaces with infinitely many lines and gives a classification of tropical lines on general smooth tropical surfaces in TP^3.
+
Title: Mass transfer through fluid-structure interactions
  
In this talk I will explain how to correct this pathology by viewing the surface as a del Pezzo cubic and considering its embedding in P^44 via its anticanonical bundle. The combinatorics of the root system of type E_6 and a tropical notion of convexity will play a central role in the construction. This is joint work in progress with Anand Deopurkar.
+
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.
  
===David Treumann (Boston College)===
+
=== William Chan (University of North Texas) ===
  
Title: Twisting things in topology and symplectic topology by pth powers
+
Title: Definable infinitary combinatorics under determinacy
  
Abstract: There's an old and popular analogy between circles and finite fields.  I'll describe some constructions you can make in Lagrangian Floer theory and in microlocal sheaf theory by taking this analogy extremely literally, the main ingredient is an "F-field."  An F-field on a manifold M is a local system of algebraically closed fields of characteristic p. When M is symplectic, maybe an F-field should remind you of a B-field, it can be used to change the Fukaya category in about the same way.  On M = S^1 times R^3, this version of the Fukaya category is related to Deligne-Lusztig theory, and I found something like a cluster structure on the Deligne-Lusztig pairing varieties by studying it.  On M = S^1 times S^1, Yanki Lekili and I have found that this version of the Fukaya category is related to the equal-characteristic version of the Fargues-Fontaine curve; the relationship is homological mirror symmetry.
+
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.
  
===Dean Baskin (Texas A&M)===
+
=== Yi Sun (Columbia) ===
  
Title: Radiation fields for wave equations
+
Title: Fluctuations for products of random matrices
  
Abstract: Radiation fields are rescaled limits of solutions of wave equations near "null infinity" and capture the radiation pattern seen by a distant observer. They are intimately connected with the Fourier and Radon transforms and with scattering theory. In this talk, I will define and discuss radiation fields in a few contexts, with an emphasis on spacetimes that look flat near infinity. The main result is a connection between the asymptotic behavior of the radiation field and a family of quantum objects on an associated asymptotically hyperbolic space.
+
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.
  
===Jianfeng Lu (Duke University)===
+
=== Zhenfu Wang (University of Pennsylvania) ===
  
Title: Density fitting: Analysis, algorithm and applications
+
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.
  
Abstract: Density fitting considers the low-rank approximation of pair products of eigenfunctions of Hamiltonian operators. It is a very useful tool with many applications in electronic structure theory. In this talk, we will discuss estimates of upper bound of the numerical rank of the pair products of eigenfunctions. We will also introduce the interpolative separable density fitting (ISDF) algorithm, which reduces the computational scaling of the low-rank approximation and can be used for efficient algorithms for electronic structure calculations. Based on joint works with Chris Sogge, Stefan Steinerberger, Kyle Thicke, and Lexing Ying.
+
===Shai Evra (IAS)===
  
===Alexei Poltoratski (Texas A&M)===
+
Title: Golden Gates in PU(n) and the Density Hypothesis.
  
Title: Completeness of exponentials: Beurling-Malliavin and type problems
+
Abstract: In their seminal work from the 80’s, Lubotzky, Phillips and Sarnak gave explicit constructions of topological generators for PU(2) with optimal covering properties. In this talk I will describe some recent works that extend the construction of LPS to higher rank compact Lie groups.
  
Abstract: This talk is devoted to two old problems of harmonic analysis mentioned in the title. Both
+
A key ingredient in the work of LPS is the Ramanujan conjecture for U(2), which follows from Deligne's proof of the Ramanujan-Petersson conjecture for GL(2). Unfortunately, the naive generalization of the Ramanujan conjecture is false for higher rank groups. Following a program initiated by Sarnak in the 90's, we prove a density hypothesis and use it as a replacement of the naive Ramanujan conjecture.
problems ask when a family of complex exponentials is complete (spans) an L^2-space. The Beruling-Malliavin
 
problem was solved in the early 1960s and I will present its classical solution along with modern generalizations
 
and applications. I will then discuss history and recent progress in the type problem, which stood open for
 
more than 70 years.
 
  
===Li-Cheng Tsai (Columbia University)===
+
This talk is based on some joint works with Ori Parzanchevski and Amitay Kamber.
  
Title: When particle systems meet PDEs
 
  
Interacting particle systems are models that involve many randomly evolving agents (i.e., particles). These systems are widely used in describing real-world phenomena. In this talk we will walk through three facets of interacting particle systems, namely the law of large numbers, random fluctuations, and large deviations. Within each facet, I will explain how Partial Differential Equations (PDEs) play a role in understanding the systems.
+
===Brett Wick (WUSTL)===
  
===Aaron Naber (Northwestern)===
+
Title: The Corona Theorem
  
Title: A structure theory for spaces with lower Ricci curvature bounds.
+
Abstract: Carleson's Corona Theorem has served as a major motivation for many results in complex function theory, operator theory and harmonic analysis. In a simple form, the result states that for $N$ bounded analytic functions $f_1,\ldots,f_N$ on the unit disc such that $\inf \left\vert f_1\right\vert+\cdots+\left\vert f_N\right\vert\geq\delta>0$ it is possible to find $N$ other bounded analytic functions $g_1,\ldots,g_N$ such that $f_1g_1+\cdots+f_Ng_N =1$. Moreover, the functions $g_1,\ldots,g_N$ can be chosen with some norm control.
  
Abstract: One should view manifolds (M^n,g) with lower Ricci curvature bounds as being those manifolds with a well behaved analysis, a point which can be rigorously stated. It thus becomes a natural question, how well behaved or badly behaved can such spaces be?  This is a nonlinear analogue to asking how degenerate can a subharmonic or plurisubharmonic function look like. In this talk we give an essentially sharp answer to this question.  The talk will require little background, and our time will be spent on understanding the basic statements and examples. The work discussed is joint with Cheeger, Jiang and with Li.
+
In this talk we will discuss some generalizations of this result to certain vector valued functions and connections with geometry and to function spaces on the unit ball in several complex variables.
 +
 
 +
===Claudia Solis Lemus===
 +
 
 +
Title New challenges in phylogenetic inference
 +
 
 +
Abstract: Phylogenetics studies the evolutionary relationships between different organisms, and its main goal is the inference of the Tree of Life. Usual statistical inference techniques like maximum likelihood and bayesian inference through Markov chain Monte Carlo (MCMC) have been widely used, but their performance deteriorates as the datasets increase in number of genes or number of species. I will present different approaches to improve the scalability of phylogenetic inference: from divide-and-conquer methods based on pseudolikelihood, to computation of Frechet means in BHV space, finally concluding with neural network models to approximate posterior distributions in tree space. The proposed methods will allow scientists to include more species into the Tree of Life, and thus complete a broader picture of evolution.
 +
 
 +
===Jessica Fintzen (Michigan)===
 +
 
 +
Title: Representations of p-adic groups
 +
 
 +
Abstract: The Langlands program is a far-reaching collection of conjectures that relate different areas of mathematics including number theory and representation theory. A fundamental problem on the representation theory side of the Langlands program is the construction of all (irreducible, smooth, complex) representations of certain matrix groups, called p-adic groups.
 +
In my talk I will introduce p-adic groups and provide an overview of our understanding of their representations, with an emphasis on recent progress. I will also briefly discuss applications to other areas, e.g. to automorphic forms and the global Langlands program.
 +
 
 +
 
 +
== Future Colloquia ==
 +
[[Colloquia/Fall 2020| Fall 2020]]
  
 
== Past Colloquia ==
 
== Past Colloquia ==
  
[[Colloquia/Blank|Blank]]
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[[Colloquia/Fall2019|Fall 2019]]
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[[Colloquia/Spring2019|Spring 2019]]
  
 
[[Colloquia/Fall2018|Fall 2018]]
 
[[Colloquia/Fall2018|Fall 2018]]
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[[Colloquia 2012-2013#Fall 2012|Fall 2012]]
 
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]
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[[WIMAW]]

Latest revision as of 20:11, 14 August 2020


Mathematics Colloquium

All colloquia are on Fridays at 4:00 pm in Van Vleck B239, unless otherwise indicated.


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) Golden Gates in PU(n) and the Density Hypothesis Gurevich
Feb 28 Brett Wick (Washington University, St. Louis) The Corona Theorem Seeger
March 6 in 911 Jessica Fintzen (Michigan) Representations of p-adic groups Marshall
March 13 CANCELLED Claudia Solis Lemus (UW-Madison, Plant Pathology) New challenges in phylogenetic inference Anderson
March 20 Spring break
March 27 CANCELLED Max Lieblich (Univ. of Washington, Seattle) Boggess, Sankar
April 3 CANCELLED Caroline Turnage-Butterbaugh (Carleton College) Marshall
April 10 No colloquium
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

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.

Shai Evra (IAS)

Title: Golden Gates in PU(n) and the Density Hypothesis.

Abstract: In their seminal work from the 80’s, Lubotzky, Phillips and Sarnak gave explicit constructions of topological generators for PU(2) with optimal covering properties. In this talk I will describe some recent works that extend the construction of LPS to higher rank compact Lie groups.

A key ingredient in the work of LPS is the Ramanujan conjecture for U(2), which follows from Deligne's proof of the Ramanujan-Petersson conjecture for GL(2). Unfortunately, the naive generalization of the Ramanujan conjecture is false for higher rank groups. Following a program initiated by Sarnak in the 90's, we prove a density hypothesis and use it as a replacement of the naive Ramanujan conjecture.

This talk is based on some joint works with Ori Parzanchevski and Amitay Kamber.


Brett Wick (WUSTL)

Title: The Corona Theorem

Abstract: Carleson's Corona Theorem has served as a major motivation for many results in complex function theory, operator theory and harmonic analysis. In a simple form, the result states that for $N$ bounded analytic functions $f_1,\ldots,f_N$ on the unit disc such that $\inf \left\vert f_1\right\vert+\cdots+\left\vert f_N\right\vert\geq\delta>0$ it is possible to find $N$ other bounded analytic functions $g_1,\ldots,g_N$ such that $f_1g_1+\cdots+f_Ng_N =1$. Moreover, the functions $g_1,\ldots,g_N$ can be chosen with some norm control.

In this talk we will discuss some generalizations of this result to certain vector valued functions and connections with geometry and to function spaces on the unit ball in several complex variables.

Claudia Solis Lemus

Title New challenges in phylogenetic inference

Abstract: Phylogenetics studies the evolutionary relationships between different organisms, and its main goal is the inference of the Tree of Life. Usual statistical inference techniques like maximum likelihood and bayesian inference through Markov chain Monte Carlo (MCMC) have been widely used, but their performance deteriorates as the datasets increase in number of genes or number of species. I will present different approaches to improve the scalability of phylogenetic inference: from divide-and-conquer methods based on pseudolikelihood, to computation of Frechet means in BHV space, finally concluding with neural network models to approximate posterior distributions in tree space. The proposed methods will allow scientists to include more species into the Tree of Life, and thus complete a broader picture of evolution.

Jessica Fintzen (Michigan)

Title: Representations of p-adic groups

Abstract: The Langlands program is a far-reaching collection of conjectures that relate different areas of mathematics including number theory and representation theory. A fundamental problem on the representation theory side of the Langlands program is the construction of all (irreducible, smooth, complex) representations of certain matrix groups, called p-adic groups. In my talk I will introduce p-adic groups and provide an overview of our understanding of their representations, with an emphasis on recent progress. I will also briefly discuss applications to other areas, e.g. to automorphic forms and the global Langlands program.


Future Colloquia

Fall 2020

Past Colloquia

Fall 2019

Spring 2019

Fall 2018

Spring 2018

Fall 2017

Spring 2017

Fall 2016

Spring 2016

Fall 2015

Spring 2015

Fall 2014

Spring 2014

Fall 2013

Spring 2013

Fall 2012

WIMAW