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	Qing Nie (UC Irvine)	TBA	Craciun
Oct 4	Su Gao (Nankai University)	TBA	Lempp
Oct 11	Mikaela Iacobelli (ETH Zurich)	TBA	Li
Oct 18	Guillaume Bal (U Chicago)	TBA	Li, Stechmann
Oct 25	Connor Mooney (UC Irvine)	TBA	Albritton
Nov 1	Dima Arinkin (UW-Madison)	TBA
Nov 4-8	Distinguished Lectures by Maksym Radziwill (Northwestern)	TBA
Nov 15	Reserved by HC	TBA	Stechmann
Nov 22	Reserved by HC	TBA	Stechmann
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.)