Difference between revisions of "Colloquia"

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= Mathematics Colloquium =
 
= Mathematics Colloquium =
  
Line 5: Line 7:
  
  
==Fall 2019==
+
==Spring 2020==
 +
 
 
{| cellpadding="8"
 
{| cellpadding="8"
 
!align="left" | date   
 
!align="left" | date   
Line 11: Line 14:
 
!align="left" | title
 
!align="left" | title
 
!align="left" | host(s)
 
!align="left" | host(s)
|-
 
|Sept 6 '''Room 911'''
 
| Will Sawin (Columbia)
 
| [[#Will Sawin (Columbia) | On Chowla's Conjecture over F_q[T] ]]
 
| Marshall
 
|-
 
|Sept 13
 
| [https://www.math.ksu.edu/~soibel/ Yan Soibelman] (Kansas State)
 
|[[#Yan Soibelman (Kansas State)|  Riemann-Hilbert correspondence and Fukaya categories ]]
 
| Caldararu
 
 
|
 
|
 
|-
 
|-
|Sept 16 '''Monday Room 911'''
+
|Jan 10
| [http://mate.dm.uba.ar/~alidick/ Alicia Dickenstein] (Buenos Aires)
+
|Thomas Lam (Michigan)  
|[[#Alicia Dickenstein (Buenos Aires)| Algebra and geometry in the study of enzymatic cascades ]]
+
|[[#Thomas Lam (Michigan) |Positive geometries and string theory amplitudes]]
| Craciun
+
|Erman
|
 
 
|-
 
|-
|Sept 20
+
|Jan 21  '''Tuesday 4-5 pm in B139'''
| [https://math.duke.edu/~jianfeng/ Jianfeng Lu] (Duke)
+
|[http://www.nd.edu/~cholak/ Peter Cholak] (Notre Dame)  
|[[#Jianfeng Lu (Duke) | How to "localize" the computation?]]
+
|[[#Peter Cholak (Notre Dame) |What can we compute from solutions to combinatorial problems?]]
| Qin
+
|Lempp
|
 
 
|-
 
|-
|Sept 26 '''Thursday 3-4 pm Room 911'''
+
|Jan 24
| [http://eugeniacheng.com/ Eugenia Cheng] (School of the Art Institute of Chicago)
+
|[https://math.duke.edu/people/saulo-orizaga Saulo Orizaga] (Duke)
| [[#Eugenia Cheng (School of the Art Institute of Chicago)| Character vs gender in mathematics and beyond ]]
+
|[[#Saulo Orizaga (Duke) | Introduction to phase field models and their efficient numerical implementation ]]
| Marshall / Friends of UW Madison Libraries
 
 
|
 
|
 
|-
 
|-
|Sept 27
+
|Jan 27 '''Monday 4-5 pm in 911'''
| Omer Mermelstein (Madison)
+
|[https://math.yale.edu/people/caglar-uyanik Caglar Uyanik] (Yale)
|
+
|[[#Caglar Uyanik (Yale) | Hausdorff dimension and gap distribution in billiards ]]
|Andrews
+
|Ellenberg
 
|-
 
|-
|Oct 4
+
|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
 
|-
 
|-
|Oct 11
+
|Jan 31
|
+
|[https://services.math.duke.edu/~pierce/ Lillian Pierce] (Duke)
|
+
|[[#Lillian Pierce (Duke) |On Bourgain’s counterexample for the Schrödinger maximal function]]
 +
|Marshall/Seeger
 
|-
 
|-
|Oct 18
+
|Feb 7
| Shamgar Gurevich (Madison)
+
|[https://web.math.princeton.edu/~jkileel/ Joe Kileel] (Princeton)
| [[#Shamgar Gurevich (Madison) | Harmonic Analysis on GL(n) over Finite Fields ]]
+
|[[#Joe Kileel (Princeton) |Inverse Problems, Imaging and Tensor Decomposition]]
| Marshall
 
|-
 
|Oct 25
 
|
 
|-
 
|Nov 1
 
|Elchanan Mossel (MIT)
 
|Distinguished Lecture
 
 
|Roch
 
|Roch
 
|-
 
|-
|Nov 8
+
|Feb 10
|Reserved for job talk
+
|[https://clvinzan.math.ncsu.edu/ Cynthia Vinzant] (NCSU)
|
+
|[[#Cynthia Vinzant (NCSU) |Matroids, log-concavity, and expanders]]
 +
|Roch/Erman
 
|-
 
|-
|Nov 15
+
|Feb 12 '''Wednesday 4-5 pm in VV 911'''
|Reserved for job talk
+
|[https://www.machuang.org/ Jinzi Mac Huang] (UCSD)
|
+
|[[#Jinzi Mac Huang (UCSD) |Mass transfer through fluid-structure interactions]]
 +
|Spagnolie
 
|-
 
|-
|Nov 22
+
|Feb 14
|Reserved for job talk
+
|[https://math.unt.edu/people/william-chan/ William Chan] (University of North Texas)
|
+
|[[#William Chan (University of North Texas) |Definable infinitary combinatorics under determinacy]]
 +
|Soskova/Lempp
 
|-
 
|-
|Nov 29
+
|Feb 17
|Thanksgiving
+
|[https://yisun.io/ Yi Sun] (Columbia)
|
+
|[[#Yi Sun (Columbia) |Fluctuations for products of random matrices]]
|-
+
|Roch
|Dec 6
 
|Reserved for job talk
 
|
 
|-
 
|Dec 11 '''Wednesday'''
 
|Nick Higham (Manchester)
 
|LAA lecture
 
|Brualdi
 
|
 
|-
 
|Dec 13
 
|Reserved for job talk
 
|
 
|}
 
 
 
==Spring 2020==
 
 
 
{| cellpadding="8"
 
!align="left" | date 
 
!align="left" | speaker
 
!align="left" | title
 
!align="left" | host(s)
 
|
 
|-
 
|Jan 24
 
|Reserved for job talk
 
|
 
|-
 
|Jan 31
 
|Reserved for job talk
 
|
 
 
|-
 
|-
|Feb 7
+
|Feb 19
|Reserved for job talk
+
|[https://www.math.upenn.edu/~zwang423// Zhenfu Wang] (University of Pennsylvania)
|
+
|[[#Zhenfu Wang (University of Pennsylvania) |Quantitative Methods for the Mean Field Limit Problem]]
|-
+
|Tran
|Feb 14
 
|Reserved for job talk
 
|
 
 
|-
 
|-
 
|Feb 21
 
|Feb 21
 +
|Shai Evra (IAS)
 +
|[[#Shai Evra (IAS) |Golden Gates in PU(n) and the Density Hypothesis]]
 +
|Gurevich
 
|
 
|
 
|-
 
|-
Line 144: Line 100:
 
|-
 
|-
 
|March 27
 
|March 27
|(Moduli Spaces Conference)
+
|[https://max.lieblich.us/ Max Lieblich] (Univ. of Washington, Seattle)
 
|
 
|
 
|Boggess, Sankar
 
|Boggess, Sankar
Line 159: Line 115:
 
|-
 
|-
 
|April 17
 
|April 17
|Song Sun (Berkeley)
+
|JM Landsberg (TAMU)
|
+
|TBA
|Huang
+
|Gurevich
 +
|-
 +
|April 23
 +
|Martin Hairer (Imperial College London)
 +
|Wolfgang Wasow Lecture
 +
|Hao Shen
 
|-
 
|-
 
|April 24
 
|April 24
Line 176: Line 137:
 
== Abstracts ==
 
== Abstracts ==
  
 +
=== Thomas Lam (Michigan) ===
  
===Will Sawin (Columbia)===
+
Title: Positive geometries and string theory amplitudes
  
Title: On Chowla's Conjecture over F_q[T]
+
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.
  
Abstract: The Mobius function in number theory is a sequences of 1s,
+
This talk is based on joint work with Arkani-Hamed, Bai, and He.
-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.
 
  
 +
=== Peter Cholak (Notre Dame) ===
  
===Yan Soibelman (Kansas State)===
+
Title: What can we compute from solutions to combinatorial problems?
  
Title: Riemann-Hilbert correspondence and Fukaya categories
+
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!
  
Abstract: In this talk I am going to discuss the role of Fukaya categories in the Riemann-Hilbert correspondence
+
=== Saulo Orizaga (Duke) ===
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.
 
  
 +
Title: Introduction to phase field models and their efficient numerical implementation
  
===Alicia Dickenstein (Buenos Aires)===
+
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.
  
Title: Algebra and geometry in the study of enzymatic cascades
+
=== Caglar Uyanik (Yale) ===
  
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.
+
Title: Hausdorff dimension and gap distribution in billiards
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.
+
                                                                                                                                             
 +
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) ===
  
=== Jianfeng Lu (Duke) ===
+
Title: Topological dynamics of countable groups and structures
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.
+
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.
  
===Eugenia Cheng (School of the Art Institute of Chicago)===
+
=== Lillian Pierce (Duke) ===
  
Title: Character vs gender in mathematics and beyond
+
Title: On Bourgain’s counterexample for the Schrödinger maximal function
  
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.
+
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) ===
  
===Shamgar Gurevich (Madison)===
+
Title: Inverse Problems, Imaging and Tensor Decomposition
  
Title: Harmonic Analysis on GL(n) over Finite Fields.
+
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.
  
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:
+
=== Cynthia Vinzant (NCSU) ===
  
trace(ρ(g)) / dim(ρ),
+
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
 
   
 
   
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.
+
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.
   
+
 
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.  
+
===Shai Evra (IAS)===
   
+
 
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.
+
Title: Golden Gates in PU(n) and the Density Hypothesis.
   
+
 
This is joint work with Roger Howe (Yale and Texas AM). The numerics for this work was carried by Steve Goldstein (Madison)
+
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.
 +
 
 +
== Future Colloquia ==
 +
[[Colloquia/Fall 2020| Fall 2020]]
  
 
== Past Colloquia ==
 
== Past Colloquia ==
  
[[Colloquia/Blank|Blank]]
+
[[Colloquia/Fall2019|Fall 2019]]
  
 
[[Colloquia/Spring2019|Spring 2019]]
 
[[Colloquia/Spring2019|Spring 2019]]

Latest revision as of 14:45, 15 February 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) 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

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.

Future Colloquia

Fall 2020

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