UW Madison mathematics Colloquium is on Fridays at 4:00 pm.

Fall 2021

September 17, 2021, Social Sciences 5208 + Live Stream, Mark Shusterman (Harvard)

(hosted by Gurevich)

Finitely Presented Groups in Arithmetic Geometry

I will report on recent works, in part joint with Esnault—Srinivas, and with Jarden, on the finite presentability of several (profinite) groups arising in algebraic geometry and in number theory. These results build on a cohomological criterion of Lubotzky involving Euler characteristics. I will try to explain the analogy, rooted in arithmetic topology, between these results and classical facts about fundamental groups of three-dimensional manifolds.

September 24, 2021, B239 + Zoom stream, Sean Paul (UW-Madison)

The Tian-Yau-Donaldson conjecture for general polarized manifolds

According to the Yau-Tian-Donaldson conjecture, the existence of a constant scalar curvature Kähler (cscK) metric in the cohomology class of an ample line bundle L on a compact complex manifold X should be equivalent to an algebro-geometric "stability condition" satisfied by the pair (X,L). The cscK metrics are the critical points of Mabuchi's K-energy functional M, defined on the space of Kähler potentials, and an important result of Chen-Cheng shows that cscK metrics exist iff M satisfies a standard growth condition (coercivity/properness). Recently the speaker has shown that the K-energy is indeed proper if and only if the polarized manifold is stable. The stability condition is closely related to the classical notion of Hilbert-Mumford stability. The speaker will give a non-technical general account of the many areas of mathematics that are involved in the proof. In particular, he hopes to discuss the surprising role played by arithmetic geometryin the spirit of Arakelov, Faltings, and Bismut-Gillet- Soule.

October 1, 2021, B239 + Live stream, Andrei Caldararu (UW-Madison)

Yet another Moonshine

The j-function, introduced by Felix Klein in 1879, is an essential ingredient in the study of elliptic curves. It is Z-periodic on the complex upper half-plane, so it admits a Fourier expansion. The original Monstrous Moonshine conjecture, due to McKay and Conway/Norton in the 1980s, relates the Fourier coefficients of the j-function around the cusp to dimensions of irreducible representations of the Monster simple group. It was proved by Borcherds in 1992.

In my talk I will try to give a rudimentary introduction to modular forms, explain Monstrous Moonshine, and discuss a new version of it obtained in joint work with Yunfan He and Shengyuan Huang. Our version involves studying the j-function around CM points (so-called Landau-Ginzburg points in the physics literature) and expanding with respect to a coordinate which arises naturally in string theory.

October 8, 2021, Zoom + live video on the 9th floor, Jon Chapman (University of Oxford)

(Wasow lecture; hosted by Thiffeault)

Asymptotics beyond all orders: the devil's invention?

"Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever." --- N. H. Abel.

The lecture will introduce the concept of an asymptotic series, showing how useful divergent series can be, despite Abel's reservations. We will then discuss Stokes' phenomenon, whereby the coefficients in the series appear to change discontinuously. We will show how understanding Stokes' phenomenon is the key which allows us to determine the qualitative and quantitative behaviour of the solution in many practical problems. Examples will be drawn from the areas of surface waves on fluids, crystal growth, dislocation dynamics, and Hele-Shaw flow.

October 11, 13, 15, 2021, Zoom, [Mon, Wed, Fri 4-5pm], Geordie Williamson (University of Sydney)

(Distinguished Lecture Series; hosted by Gurevich)

Geometric representation theory and modular representations

Representation theory is the study of linear symmetry. We are interested in all ways in which a group can arise as the symmetries of a vector space. Representation theory is a remarkably rich subject, with deep connections to number theory, combinatorics, algebraic geometry, differential geometry, theoretical physics and beyond. This lecture series will focus on modular representations, i.e. those representations where our vector spaces are over a field of characteristic p. I will try to highlight some of the main questions in the field and why we are interested in answering them. It is remarkable how much is still unknown and how hard some of these questions are. I will explain the role played by geometric representation theory in our attempts to understand these questions. A fascinating blend of algebra, algebraic geometry, category theory and algebraic topology is informing our understanding of basic questions. Much remains to be understood!

October 22, 2021, Zoom, Vera Serganova (UC Berkeley)

(hosted by Gurevich/Gorin)

Supersymmetry and tensor categories

I will explain how representation theory of supergroups and supergeometry are related to general theory of tensor categories, present old and new results and open questions in the field. We will see how universal tensor categories can be constructed using supergroups and discuss analogy between super representation theory and representation theory over the fields of positive characteristic.

October 29, 2021, Zoom, Alexandru Ionescu (Princeton University)

(hosted by Wainger)

Polynomial averages and pointwise ergodic theorems on nilpotent groups

I will talk about some recent work on pointwise almost everywhere convergence for ergodic averages along polynomial sequences in nilpotent groups of step two. Our proof is based on almost-orthogonality techniques that go far beyond Fourier transform tools, which are not available in the non-commutative nilpotent setting. In particular we develop what we call a nilpotent circle method}, which allows us to adapt some the ideas of the classical circle method to the setting of nilpotent groups.

November 5, 2021, B239 + Live stream, Jayadev S. Athreya (University of Washington)

(hosted by Uyanik)

Surfaces and Point Processes

We'll give several concrete examples of how to go from the geometry of surfaces to the study of point processes, following work of Siegel, Veech, Masur, Eskin, Mirzakhani, Wright, and others. We'll discuss how this "probabilistic" perspective helps inform both the direction of questions one asks, as well as providing ideas of how to prove things. We'll discuss some pieces of joint work with Cheung-Masur, Margulis, and Arana-Herrera.

November 12, 2021, Zoom, Kasso Okoudjou (Tufts University)

(hosted by Stovall)

An exploration in analysis on fractals

Analysis on fractal sets such as the Sierpinski gasket is based on the spectral analysis of a corresponding Laplace operator. In the first part of the talk, I will describe a class of fractals and the analytical tools that they support. In the second part of the talk, I will consider fractal analogs of topics from classical analysis, including the Heisenberg uncertainty principle, the spectral theory of Schrödinger operators, and the theory of orthogonal polynomials.

November 19, 2021 , B239 + Zoom, Albert Ai(UW-Madison)

(reserved by the hiring committee)

Low regularity solution for quasilinear PDEs

In this talk, we will consider the low regularity well-posedness problem for a pair of quasilinear dispersive PDEs: the nonlinear wave equation, and the water waves equations. Two classical methods, energy estimates and Strichartz estimates, have historically yielded substantial but partial results toward advancing the low regularity theory. We will see how, using a special structure of the equations known as a normal form structure, combined with tools from harmonic and microlocal analysis, we can refine these classical methods to drastically improve the known results for low regularity well-posedness.