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 + Zoom 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, 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.