Mathematics Colloquium

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

Fall 2019

date	speaker	title	host(s)
Sept 6 Room 911	Will Sawin (Columbia)	On Chowla's Conjecture over F_q[T]	Marshall
Sept 13	Yan Soibelman (Kansas State)	Riemann-Hilbert correspondence and Fukaya categories	Caldararu
Sept 16 Monday Room 911	Alicia Dickenstein (Buenos Aires)	Algebra and geometry in the study of enzymatic cascades	Craciun
Sept 20	Jianfeng Lu (Duke)	TBA	Qin
Sept 27	Omer Mermelstein (Madison)		Andrews
Oct 4
Oct 11
Oct 18	Shamgar Gurevich (Madison)	Harmonic Analysis on GL(n) over Finite Fields	Marshall
Oct 25
Nov 1	Elchanan Mossel (MIT)	Distinguished Lecture	Roch
Nov 8	Reserved for job talk
Nov 15	Reserved for job talk
Nov 22	Reserved for job talk
Nov 29	Thanksgiving
Dec 6	Reserved for job talk
Dec 11 Wednesday	Nick Higham (Manchester)	LAA lecture	Brualdi
Dec 13	Reserved for job talk

Spring 2020

date	speaker	title	host(s)
Jan 24	Reserved for job talk
Jan 31	Reserved for job talk
Feb 7	Reserved for job talk
Feb 14	Reserved for job talk
Feb 21
Feb 28	Brett Wick (Washington University, St. Louis)		Seeger
March 6	Jessica Fintzen (Michigan)		Marshall
March 13
March 20	Spring break
March 27	(Moduli Spaces Conference)		Boggess, Sankar
April 3	Caroline Turnage-Butterbaugh (Carleton College)		Marshall
April 10	Sarah Koch (Michigan)		Bruce (WIMAW)
April 17
April 24	Natasa Sesum (Rutgers University)		Angenent
May 1	Robert Lazarsfeld (Stony Brook)	Distinguished lecture	Erman

Abstracts

Will Sawin (Columbia)

Title: On Chowla's Conjecture over F_q[T]

Abstract: The Mobius function in number theory is a sequences of 1s, -1s, and 0s, which is simple to define and closely related to the prime numbers. Its behavior seems highly random. Chowla's conjecture is one precise formalization of this randomness, and has seen recent work by Matomaki, Radziwill, Tao, and Teravainen making progress on it. In joint work with Mark Shusterman, we modify this conjecture by replacing the natural numbers parameterizing this sequence with polynomials over a finite field. Under mild conditions on the finite field, we are able to prove a strong form of this conjecture. The proof is based on taking a geometric perspective on the problem, and succeeds because we are able to simplify the geometry using a trick based on the strange properties of polynomial derivatives over finite fields.

Yan Soibelman (Kansas State)

Title: Riemann-Hilbert correspondence and Fukaya categories

Abstract: In this talk I am going to discuss the role of Fukaya categories in the Riemann-Hilbert correspondence for differential, q-difference and elliptic difference equations in dimension one. This approach not only gives a unified answer for several versions of the Riemann-Hilbert correspondence but also leads to a natural formulation of the non-abelian Hodge theory in dimension one. It also explains why periodic monopoles should appear as harmonic objects in this generalized non-abelian Hodge theory. All that is a part of the bigger project ``Holomorphic Floer theory", joint with Maxim Kontsevich.

Alicia Dickenstein (Buenos Aires)

Title: Algebra and geometry in the study of enzymatic cascades

Abstract: In recent years, techniques from computational and real algebraic geometry have been successfully used to address mathematical challenges in systems biology. The algebraic theory of chemical reaction systems aims to understand their dynamic behavior by taking advantage of the inherent algebraic structure in the kinetic equations, and does not need the determination of the parameters a priori, which can be theoretically or practically impossible. I will give a gentle introduction to general results based on the network structure. In particular, I will describe a general framework for biological systems, called MESSI systems, that describe Modifications of type Enzyme-Substrate or Swap with Intermediates, and include many networks that model post-translational modifications of proteins inside the cell. I will also outline recent methods to address the important question of multistationarity, in particular in the study of enzymatic cascades, and will point out some of the mathematical challenges that arise from this application.

Shamgar Gurevich (Madison)

Title: Harmonic Analysis on GL(n) over Finite Fields.

Abstract: There are many formulas that express interesting properties of a finite group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the character ratio:

trace(ρ(g)) / dim(ρ),

for an irreducible representation ρ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of the mentioned type for analyzing certain random walks on G.

Recently, we discovered that for classical groups G over finite fields there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant rank.

This talk will discuss the notion of rank for the group GLn over finite fields, demonstrate how it controls the character ratio, and explain how one can apply the results to verify mixing time and rate for certain random walks.

This is joint work with Roger Howe (Yale and Texas AM). The numerics for this work was carried by Steve Goldstein (Madison)