Colloquia/Fall 2024

Organizers: Dallas Albritton and Michael Kemeny

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

mathcolloquium@g-groups.wisc.edu is the mailing list. Everyone in the math department is subscribed.

date	speaker	title	host(s)
Sept 6	Dan Romik (UC Davis)	Sphere packing in dimension 8, Viazovska's solution, and a new human proof of her modular form inequalities	Gurevitch
Sept 13	No Colloquium
Sept 20	Alireza Golsefidy (UCSD)	Closure of orbits of the pure mapping class group on the character variety	Marshall
Sept 25 (Wednesday) in VV 911 (not in B239)	Qing Nie (UC Irvine)	Systems Learning of Single Cells	Craciun
Oct 4	Su Gao (Nankai University)	Continuous combinatorics of countable abelian group actions	Lempp
Oct 11 in VV 911 (not in B239)	Mikaela Iacobelli (ETH Zurich/IAS Princeton)	Challenges and Breakthroughs in the Mathematics of Plasmas	Li
Oct 18	Guillaume Bal (U Chicago)	Speckle formation of laser light in random media: The Gaussian conjecture	Li, Stechmann
Oct 25	Connor Mooney (UC Irvine)	Minimal graphs in higher codimension	Albritton
Nov 1	Dima Arinkin (UW-Madison)	What is the geometric Langlands conjecture?
Nov 5 (Tuesday), 4-5 pm in Sterling 1313 Nov 6 (Wednesday), 2-3 pm in Sterling 2335 Nov 7 (Thursday), 4-5 pm in VV 911	Maksym Radziwill (Northwestern)	Distinguished Lecture Series	Guo
Nov 15	Matthew Stover (Temple)	Negative curvature, branched covers, and central extensions	Stechmann
Nov 22	Peter Morfe (Max Planck Institute, Leipzig)	TBA	Feldman
Nov 29	Thanksgiving holiday break
Dec 6	Reserved by HC	TBA	Stechmann
Dec 13	Reserved by HC	TBA	Stechmann

Abstracts

September 6: Dan Romik (UC Davis)

Title: Sphere packing in dimension 8, Viazovska's solution, and a new human proof of her modular form inequalities

Abstract: Maryna Viazovska in 2016 found a remarkable application of complex analysis and the theory of modular forms to a fundamental problem in geometry, obtaining a solution to the sphere packing problem in dimension 8 through an explicit construction of a so-called "magic function" that she defined in terms of classical special functions. The same method also led shortly afterwards to the solution of the sphere packing problem in dimension 24 by her and several collaborators. One component of Viazovska's proof consisted of proving a pair of inequalities satisfied by the modular forms she constructed. Viazovska gave a proof of these inequalities that relied in an essential way on computer calculations. In this talk I will describe the background leading up to Viazovska's groundbreaking proof, and present a new proof of her inequalities that uses only elementary arguments that can be easily checked by a human.

September 20: Alireza Golsefidy (UCSD)

Closure of orbits of the pure mapping class group on the character variety

For every surface S, the pure mapping class group G_S acts on the (SL_2)-character variety Ch_S of a fundamental group P of S. The character variety Ch_S is a scheme over the ring of integers. Classically this action on the real points Ch_S(R) of the character variety has been studied in the context of the Teichmuller theory and SL(2,R)-representations of P. In a seminal work, Goldman studied this action on a subset of Ch_S(R) which comes from SU(2)-representations of P. In this case, Goldman showed that if S is of genus g>1 and zero punctures, then the action of G_S is ergodic. Previte and Xia studied this question from topological point of view, and when g>0, proved that the orbit closure is as large as algebraically possible. Bourgain, Gamburd, and Sarnak studied this action on the F_p-points Ch_S(F_p) of the character variety where S is a puncture torus. They conjectured that in this case, this action has only two orbits, where one of the orbits has only one point. Recently, this conjecture was proved for large enough primes by Chen. When S is an n-punture sphere, the finite orbits of this action on Ch_S(C) are connected to the algebraic solutions of Painleve differential equations. In this talk, I will report on our recent contributions to this theory. Here are some sample results:

• An almost complete description of the Zariski-closure of infinite G_S-orbits in Ch_S(F) where F is a characteristic zero field.

• Answering a question of Goldman-Previte-Xia by understanding the orbit closure of G_S on SU(2)-representation part of Ch_S(R) where S is an n-puncture sphere.

• Show that the original result of Previte and Xia is not accurate and give a description of the cases where it fails.

• Proving that in most cases the closure of G_S-orbits in the p-adic integer points Ch_S(Z_p) are open within given polynomial constrains. We give precise description of exceptional cases.

(This is a joint work with Natallie Tamam.)

September 25: Qing Nie (UC Irvine)

Title: Systems Learning of Single Cells

Cells make fate decisions in response to dynamic environments, and multicellular structures emerge from multiscale interplays among cells and genes in space and time. The recent single-cell genomics technology provides an unprecedented opportunity to profile cells for all their genes. While those measurements provide high-dimensional gene expression profiles for all cells, it requires fixing individual cells that lose many important spatiotemporal information. Is it possible to infer temporal relationships among cells from single or multiple snapshots? How to recover spatial interactions among cells, for example, cell-cell communication? In this talk I will present our newly developed computational tools to study cell fate in the context of single cells as a system. In particular, I will show dynamical models and machine-learning methods, with a focus on inference and analysis of transitional properties of cells and cell-cell communication using both high-dimensional single-cell and spatial transcriptomics, as well as multi-omics data for some cases. Through their applications to various complex systems in development, regeneration, and diseases, we show the discovery power of such methods in addition to identifying areas for further method development for spatiotemporal analysis of single-cell data.

October 4: Su Gao (Nankai University)

Title: Continuous combinatorics of countable abelian group actions

Abstract: We consider combinatorial problems on the free parts of the Bernoulli shift actions of countable abelian groups, such as chromatic numbers, edge chromatic numbers, perfect matchings, etc. These problems can all be regarded as special cases of the problem whether there exist continuous equivariant maps from the free part of the Bernoulli shift action to a subshift of finite type. We prove a master theorem which in theory gives complete answers to the subshift problem. Furthermore, we show that the class of (codes for) all subshifts of finite type with a positive answer to the subshift problem is a complete c.e. set. This is joint work with Steve Jackson, Ed Krohne, and Brandon Seward.

October 11: Mikaela Iacobelli (ETH/IAS)

Title: Challenges and Breakthroughs in the Mathematics of Plasmas

Abstract: This colloquium will explore some fundamental issues in the mathematics of plasmas, focusing on the stability and instability of solutions to Vlasov-type equations, which are crucial for describing the behavior of charged particles in a plasma. A general introduction to kinetic theory is given, making the subject accessible to a wide audience of mathematicians. Key mathematical concepts such as well-posedness, stability, and the behavior of solutions in singular limits are discussed. In addition, a new class of Wasserstein-type distances is introduced, offering new perspectives on the stability of kinetic equations.

October 18: Guillaume Bal (Chicago)

Title: Speckle formation of laser light in random media: The Gaussian conjecture

Abstract: A widely accepted conjecture in the physical literature states that classical wave-fields propagating in random media over large distances eventually follow a complex circular Gaussian distribution. In this limit, the wave intensity becomes exponentially distributed, which corroborates the speckle patterns of, e.g., laser light observed in experiments. This talk reports on recent results settling the conjecture in the weak-coupling, paraxial regime of wave propagation. The limiting macroscopic Gaussian wave-field is fully characterized by a correlation function that satisfies an unusual diffusion equation.

The paraxial model of wave propagation is an approximation of the Helmholtz model where backscattering has been neglected. It is mathematically simpler to analyze but quite accurate practically for wave-fields that maintain a beam-like structure as in the application of laser light propagating in turbulent atmospheres.

The derivation of the limiting model is first obtained in the Itô-Schrödinger regime, where the random medium is replaced by its white noise limit. The resulting stochastic PDE has the main advantage that finite dimensional statistical moments of the wave-field satisfy closed form equations. The proof of the derivation of the macroscopic model is based on showing that these moment solutions are asymptotically those of the Gaussian limit, on obtaining a stochastic continuity (and tightness) result, and on establishing that moments in the paraxial and the Itô-Schrödinger regimes are asymptotically close.

This is joint work with Anjali Nair.

October 25: Connor Mooney (UC Irvine)

Title: Minimal graphs in higher codimension

Abstract: Minimal graphs have been studied since Lagrange wrote the minimal surface equation around 1760. By the 1970s, many beautiful results had been proven for the minimal surface equation (existence, regularity, Bernstein theorem). Around that time, Lawson and Osserman showed that most of these results are false for the minimal surface system. However, there are conditions under which such results can be extended to the minimal surface system, and we've recently answered some fundamental questions about this system that the work of Lawson and Osserman left open. I will discuss a few of these advances.

November 1: Dima Arinkin (UW Madison)

Title: What is the geometric Langlands conjecture?

Abstract: The Langlands program originated in a series of conjectures formulated by Robert Langlands in late 1960's. A geometric version of the conjectures relates two natural spaces associated to a Riemann surface: the space of vector bundles and the space of local systems. The geometric Langlands program went through several transformations and grew to connect many areas of mathematics.

My goal is to provide an informal introduction to the (global) geometric Langlands conjecture, and sketch some recent developments, combining classical ideas and modern tools.

Distinguished Lectures by Maksym Radziwill (Northwestern)

Lecture 1: Tuesday (Nov 5), 4-5pm – Sterling 1313

Lecture 2: Wednesday (Nov 6), 2-3pm – Sterling 2335

Lecture 3: Thursday (Nov 7), 4-5pm – Van Vleck 911

Lecture 1: I will introduce some of the basic properties of the Riemann zeta-function and discuss their arithmetic meaning. I will then explain how these basic properties lead to the Riemann Hypothesis and its significance. Finally I will explain in what circumstances we expect analogues of the Riemann Hypothesis to be true and what happens when we don't expect such analogues to hold.

Lecture 2: I will start by discussing Selberg's central limit theorem, which describes the typical behavior of the Riemann zeta-function. An extrapolation of this result leads to conjectures about global and local maxima of the Riemann zeta-function, which are however wrong in subtle ways. I will then discuss ways in which these predictions can be corrected. This will lead us into a discussion of the random matrix theory model for the Riemann zeta-function (and how it came to be) and a discussion of the recently discovered branching structure (as in "branching random walks") in the Riemann zeta-function.

Lecture 3: I will discuss recent work (joint with Arguin and Bourgade) in which we settle the Fyodorov-Hiary-Keating conjecture concerning the local maxima of the Riemann zeta-function. Based on the second lecture, this conjecture also admits a natural variant describing the maxima of characteristic polynomials of random unitary matrices. The latter has a physical significance that I will explain. I will then explain the broad ideas that appear in the proof of this conjecture and the connections with branching random walks. Time permitting, I will explain the connection with some techniques arising in sieve theory

November 15, 2024 Matthew Stover (Temple)

By the classical uniformization theorem, a smooth complex projective curve admits a Riemannian metric of constant curvature $-1$ if and only if the genus is at least two. Mostow and Siu famously proved in 1980 that the analogous problem for higher-dimensional Kähler manifolds has a negative solution: there are $2$-dimensional compact complex manifolds with a negatively curved Kähler metric that do not admit a metric of constant holomorphic sectional curvature. Domingo Toledo and I recently extended this to all higher dimensions by a robust branched cover construction reminiscent of Gromov and Thurston's construction of negatively curved Riemannian $n$-manifolds for each $n \ge 4$ that admit no constant curvature $-1$ metric. It ends up that the key result on the way to our examples is about something called residual finiteness for central extensions of certain fundamental groups. This talk will be a general introduction of the problem of building negatively curved manifolds with no locally symmetric metric, why something like residual finiteness becomes relevant for building branched covers, and a brief overview of how deep results on arithmetic groups provide just the right starting input.