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= Mathematics Colloquium =
 
= Mathematics Colloquium =
  
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==Fall 2019==
+
==Spring 2020==
 +
 
 
{| cellpadding="8"
 
{| cellpadding="8"
 
!align="left" | date   
 
!align="left" | date   
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!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'''
|
+
|[https://math.yale.edu/people/caglar-uyanik Caglar Uyanik] (Yale)
|
+
|[[#Caglar Uyanik (Yale) | Hausdorff dimension and gap distribution in billiards ]]
 +
|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
| Omer Mermelstein (Madison)
+
|[https://services.math.duke.edu/~pierce/ Lillian Pierce] (Duke)
| [[#Omer Mermelstein (Madison)| Generic flat pregeometries ]]
+
|[[#Lillian Pierce (Duke) |On Bourgain’s counterexample for the Schrödinger maximal function]]
|Andrews
+
|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
|Jose Rodriguez (UW-Madison)
+
|[https://clvinzan.math.ncsu.edu/ Cynthia Vinzant] (NCSU)
|[[#Jose Rodriguez (UW-Madison) | Nearest Point Problems and Euclidean Distance Degrees]]
+
|[[#Cynthia Vinzant (NCSU) |Matroids, log-concavity, and expanders]]
|Erman
+
|Roch/Erman
|-
 
|Nov 15
 
|Reserved for job talk
 
 
|-
 
|-
|Nov 20 '''Wednesday 4-5pm'''
+
|Feb 12 '''Wednesday 4-5 pm in VV 911'''
|Franca Hoffman (Caltech)
+
|[https://www.machuang.org/ Jinzi Mac Huang] (UCSD)
|[[#Franca Hoffman (Caltech) | Gradient Flows: From PDE to Data Analysis]]
+
|[[#Jinzi Mac Huang (UCSD) |Mass transfer through fluid-structure interactions]]
|Smith
+
|Spagnolie
 
|-
 
|-
|Nov 22
+
|Feb 14
| Jeffrey Danciger (UT Austin)
+
|[https://math.unt.edu/people/william-chan/ William Chan] (University of North Texas)
| [[#Jeffrey Danciger (UT Austin) | "TBA"]]
+
|[[#William Chan (University of North Texas) |Definable infinitary combinatorics under determinacy]]
| Kent
+
|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)
|
+
|[[#Shai Evra (IAS) |Golden Gates in PU(n) and the Density Hypothesis]]
 
|Gurevich
 
|Gurevich
 
|
 
|
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|Feb 28
 
|Feb 28
 
|Brett Wick (Washington University, St. Louis)
 
|Brett Wick (Washington University, St. Louis)
|
+
|[[#Brett Wick (WUSTL) |The Corona Theorem]]
 
|Seeger
 
|Seeger
 
|-
 
|-
|March 6
+
|March 6 '''in 911'''
 
| Jessica Fintzen (Michigan)
 
| Jessica Fintzen (Michigan)
|
+
|[[#Jessica Fintzen (Michigan) | Representations of p-adic groups]]
 
|Marshall
 
|Marshall
 
|-
 
|-
|March 13
+
|March 13 '''CANCELLED'''
|
+
| [https://plantpath.wisc.edu/claudia-solis-lemus// Claudia Solis Lemus]  (UW-Madison, Plant Pathology)
 +
|[[#Claudia Solis Lemus | New challenges in phylogenetic inference]]
 +
|Anderson
 
|-
 
|-
 
|March 20
 
|March 20
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|
 
|
 
|-
 
|-
|March 27
+
|March 27 '''CANCELLED'''
|(Moduli Spaces Conference)
+
|[https://max.lieblich.us/ Max Lieblich] (Univ. of Washington, Seattle)
 
|
 
|
 
|Boggess, Sankar
 
|Boggess, Sankar
 
|-
 
|-
|April 3
+
|April 3 '''CANCELLED'''
 
|Caroline Turnage-Butterbaugh (Carleton College)
 
|Caroline Turnage-Butterbaugh (Carleton College)
 
|
 
|
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|-
 
|-
 
|April 10
 
|April 10
| Sarah Koch (Michigan)
+
| No colloquium
 
|
 
|
| Bruce (WIMAW)
+
|  
 
|-
 
|-
 
|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 186: Line 139:
 
== Abstracts ==
 
== 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) ===
  
===Will Sawin (Columbia)===
+
Title: Introduction to phase field models and their efficient numerical implementation
  
Title: On Chowla's Conjecture over F_q[T]
+
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.
  
Abstract: The Mobius function in number theory is a sequences of 1s,
+
=== Caglar Uyanik (Yale) ===
-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.
 
  
 +
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.
  
===Yan Soibelman (Kansas State)===
+
=== Andy Zucker (Lyon) ===
  
Title: Riemann-Hilbert correspondence and Fukaya categories
+
Title: Topological dynamics of countable groups and structures
  
Abstract: In this talk I am going to discuss the role of Fukaya categories in the Riemann-Hilbert correspondence
+
Abstract: We give an introduction to the abstract topological dynamics
for differential, q-difference and elliptic difference equations in dimension one.
+
of topological groups, i.e. the study of the continuous actions of a
This approach not only gives a unified answer for several versions of the Riemann-Hilbert correspondence but also leads to a natural formulation
+
topological group on a compact space. We are particularly interested
of the non-abelian Hodge theory in dimension one. It also explains why periodic monopoles
+
in the minimal actions, those for which every orbit is dense.
should appear as harmonic objects in this generalized non-abelian Hodge theory.
+
The study of minimal actions is aided by a classical theorem of Ellis,
All that is a part of the bigger project ``Holomorphic Floer theory",
+
who proved that for any topological group G, there exists a universal
joint with Maxim Kontsevich.
+
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) ===
  
===Alicia Dickenstein (Buenos Aires)===
+
Title: On Bourgain’s counterexample for the Schrödinger maximal function
  
Title: Algebra and geometry in the study of enzymatic cascades
+
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.”
  
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.
+
=== Joe Kileel (Princeton) ===
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.
 
  
 +
Title: Inverse Problems, Imaging and Tensor Decomposition
  
=== Jianfeng Lu (Duke) ===
+
Abstract: Perspectives from computational algebra and optimization are brought
Title: How to ``localize" the computation?
+
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.
  
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.
+
=== Cynthia Vinzant (NCSU) ===
  
===Eugenia Cheng (School of the Art Institute of Chicago)===
+
Title: Matroids, log-concavity, and expanders
  
Title: Character vs gender in mathematics and beyond
+
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.
  
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.
+
=== Jinzi Mac Huang (UCSD) ===
  
===Omer Mermelstein (Madison)===
+
Title: Mass transfer through fluid-structure interactions
  
Title: Generic flat pregeometries
+
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: 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.
+
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.
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.
 
  
 +
===Shai Evra (IAS)===
  
===Shamgar Gurevich (Madison)===
+
Title: Golden Gates in PU(n) and the Density Hypothesis.
  
Title: Harmonic Analysis on GL(n) over Finite Fields.
+
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: 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:
+
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.
  
trace(ρ(g)) / dim(ρ),
+
This talk is based on some joint works with Ori Parzanchevski and Amitay Kamber.
+
 
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.
+
 
   
+
===Brett Wick (WUSTL)===
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.  
+
 
   
+
Title: The Corona Theorem
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)
 
  
 +
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.
  
===Jose Rodriguez (UW-Madison)===
+
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.
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.
 
  
 +
===Claudia Solis Lemus===
  
===Ananth Shankar (MIT)===
+
Title New challenges in phylogenetic inference
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 surfa​ces. 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)===
+
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.
  
Title: Gradient Flows: From PDE to Data Analysis.
+
===Jessica Fintzen (Michigan)===
  
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.
+
Title: Representations of p-adic groups
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) ===
+
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.
  
Title: TBA
 
  
 
== Future Colloquia ==
 
== Future Colloquia ==
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== Past Colloquia ==
 
== Past Colloquia ==
  
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[[Colloquia/Fall2019|Fall 2019]]
  
 
[[Colloquia/Spring2019|Spring 2019]]
 
[[Colloquia/Spring2019|Spring 2019]]
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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 21:11, 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