Colloquia

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UW Madison mathematics Colloquium is on Fridays at 4:00 pm in Van Vleck B239 unless otherwise noted.

Contacts for the colloquium are Simon Marshall and Dallas Albritton.


Spring 2024

date speaker title host(s)
Monday Jan 22 at 4pm in B239 Yingkun Li (Darmstadt Tech U, Germany) Arithmetic of real-analytic modular forms Yang
Thursday Jan 25 at 4pm in VV911 Sanjukta Krishnagopal (UCLA/UC Berkeley) Theoretical methods for data-driven complex systems: from mathematical machine learning to simplicial complexes Smith
Jan 26 Jacob Bedrossian (UCLA) Lyapunov exponents in stochastic systems Tran
Feb 2 William Chen Orbit problems and the mod p properties of Markoff numbers Arinkin
Feb 9 No colloquium
Feb 16 Jack Lutz (Iowa State) Algorithmic Fractal Dimensions Guo
Feb 23
Mar 1 Per-Gunnar Martinsson (UT-Austin) Randomized algorithms for linear algebraic computations Li
Mar 8 Anton Izosimov (U of Arizona) Gloria Mari-Beffa
Mar 15 Peter Humphries (Virginia) Marshall
Mar 20 Wanlin Li (Washington U St Louis) Dymarz, GmMaW
Mar 29 Spring break
Apr 5 Ovidiu Savin (Columbia) Tran
Apr 12 Mikayla Kelley (U Chicago Philosophy) Math And... seminar, title TBA Ellenberg, Marshall
Apr 19 Yanyan Li (Rutgers) Tran
Apr 26 Chris Leininger (Rice) TBA Uyanik

Abstracts

Monday, January 22. Yingkun Li

Arithmetic of real-analytic modular forms

Modular form is a classical mathematical object dating back to the 19th century. Because of its connections to and appearances in many different areas of math and physics, it remains a popular subject today. Since the work of Hans Maass in 1949, real-analytic modular form has found important applications in arithmetic geometry and number theory. In this talk, I will discuss the amazing works in this area over the past 20 years, and give a glimpse of its fascinating future directions.

Thursday, January 25. Sanjukta Krishnagopal

Theoretical methods for data-driven complex systems: from mathematical machine learning to simplicial complexes

In this talk I will discuss some aspects at the intersection of mathematics, machine learning, and networks to introduce interdisciplinary methods with wide application.

First, I will discuss some recent advances in mathematical machine learning for prediction on graphs. Machine learning is often a black box. Here I will present some exact theoretical results on the dynamics of weights while training graph neural networks using graphons - a graph limit or a graph with infinitely many nodes. I will use these ideas to present a new method for predictive and personalized medicine applications with remarkable success in prediction of Parkinson's subtype five years in advance.

Then, I will discuss some work on higher-order models of graphs: simplicial complexes - that can capture simultaneous many-body interactions. I will present some recent results on spectral theory of simplicial complexes, as well as introduce a mathematical framework for studying the topology and dynamics of multilayer simplicial complexes using Hodge theory, and discuss applications of such interdisciplinary methods to studying bias in society, opinion dynamics, and hate speech in social media.


Friday, January 26. Jacob Bedrossian

Lyapunov exponents in stochastic systems

In this overview talk we discuss several results regarding positive Lyapunov exponents in stochastic systems. First we discuss proving "Lagrangian chaos" in stochastic fluid mechanics, that is, demonstrating a positive Lyapunov exponent for the motion of a particle in the velocity field arising from the stochastic Navier-Stokes equations. We describe how this chaos can be used to deduce qualitatively optimal almost-sure exponential mixing of passive scalars. Next we describe more recently developed methods for obtaining strictly positive lower bounds and some quantitative estimates on the top Lyapunov exponent of weakly-damped stochastic differential equations, such as Lorenz-96 model or Galerkin truncations of the 2d Navier-Stokes equations (called "Eulerian chaos" in fluid mechanics). Further applications of the ideas to the chaotic motion of charged particles in fluctuating magnetic fields and the non-uniqueness of stationary measures for Lorenz 96 in degenerate forcing situations will be discussed if time permits. All of the work except for the charged particles (joint with Chi-Hao Wu) is joint with Alex Blumenthal and Sam Punshon-Smith.

Friday, February 2. William Chen

Orbit problems and the mod p properties of Markoff numbers

Markoff numbers are positive integers which encode how resistant certain irrational numbers are to being approximated by rationals. In 1913, Frobenius asked for a description of all congruence conditions satisfied by Markoff numbers modulo primes p. In 1991 and 2016, Baragar, Bourgain, Gamburd, and Sarnak conjectured a refinement of Frobenius’s question, which amounts to showing that the Markoff equation x^2 + y^2 + z^2 - xyz = 0 satisfies “strong approximation”; that is to say: they conjecture that its integral points surject onto its mod p points for every prime p. In this talk we will show how to prove this conjecture for all but finitely many primes p, thus reducing the conjecture to a finite computation. A key step is to understand this problem in the context of describing the orbits of certain group actions. Primarily, we will consider the action of the mapping class group of a topological surface S on (a) the set of G-covers of S, where G is a finite group, and (b) on the character variety of local systems on S. Questions of this type have been related to many classical problems, from proving that the moduli space of curves of a given genus is connected, to Grothendieck’s ambitious plan to understand the structure of the absolute Galois group of the rationals by studying its action on “dessins d’enfant”. We will explain some of this history and why such problems can be surprisingly difficult.


Friday, February 16. Jack Lutz

Algorithmic Fractal Dimensions

Algorithmic fractal dimensions are computability theoretic versions of Hausdorff dimension and other fractal dimensions. This talk will introduce algorithmic fractal dimensions with particular focus on the Point-to-Set Principle. This principle has enabled several recent proofs of new theorems in geometric measure theory. These theorems, some solving long-standing open problems, are classical (meaning that their statements do not involve computability or logic), even though computability has played a central in their proofs.



Friday, March 1. Per-Gunnar Martinsson

Randomized algorithms for linear algebraic computations

The talk will describe how randomized algorithms can effectively, accurately, and reliably solve linear algebraic problems that are omnipresent in scientific computing and in data analysis. We will focus on techniques for low rank approximation, since these methods are particularly simple and powerful, and are well understood mathematically. The talk will also briefly survey a number of other randomized algorithms for tasks such as solving linear systems, estimating matrix norms, and computing full matrix factorizations.

Past Colloquia

Spring 2024

Fall 2023

Spring 2023

Fall 2022

Spring 2022

Fall 2021

Spring 2021

Fall 2020

Spring 2020

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

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