UMN Topology Seminar

Configuration spaces and applications in arithmetic statistics

Anh Trong Nam Hoang , University of Minnesota

In the last dozen years, topological methods have been shown to produce a new pathway to study arithmetic statistics over function fields, most notably in Ellenberg-Venkatesh-Westerland's work on the Cohen-Lenstra conjecture. More recently, Ellenberg, Tran and Westerland proved the upper bound in Malle’s conjecture for function fields by studying stability of the homology of braid groups with certain exponential coefficients. In this talk, we will give an overview of their framework and extend their techniques to study other questions in arithmetic statistics. As an example, we will demonstrate how this extension can be used to study character sums of the resultant of monic square-free polynomials over finite fields, answering and generalizing a question of Ellenberg and Shusterman.

VH 570 Zoom link: https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327

The Alexander trick for homology spheres.

Søren Galatius ,University of Copenhagen

It is well known that the homeomorphism group of a disk relative to its boundary is contractible. This is known as the Alexander trick, and was published 100 years ago. I will discuss joint work with Randal-Williams on the homeomorphism group of a compact contractible manifold relative to its (not necessarily simply connected) boundary, which we prove to be contractible if the dimension is at least 6.

VH 570 Zoom link: https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327

The exceptional symmetry of loop spaces

Sasha Voronov, University of Minnesota

The quest we have started with Hisham Sati for physically motivated structures on certain loop spaces continues. In this talk I will describe our latest finds: the identification of the rational homotopy type (represented by a minimal model) of the k-fold free loop space divided by the k-torus T^k action, Map(T^k, S^4)//T^k and finding the E_k-symmetry of that minimal model, where E_k stands for the exceptional series of simple Lie algebras.Location: VH 570 Zoom link: https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327

Mackey and Tambara Functors Beyond Equivariant Homotopy
Abstract: "Classically", Mackey and Tambara functors are equivariant generalizations of abelian groups and commutative rings, respectively. What this means is that, in equivariant homotopy theory, Mackey functors appear wherever one would expect to find abelian groups, and Tambara functors appear wherever one would expect to find commutative rings. More recently, work by Bachmann has garnered interest in related structures which appear in motivic homotopy theory -- these Motivic Mackey Functors and Motivic Tambara Functors do not have anything to do with group-equivariance, but have the same axiomatic. In this talk, I'll introduce a general context for interpreting the notions of Mackey and Tambara Functors, which subsumes both the equivariant and motivic notions. The aim of this approach is to translate theorems between contexts, enriching the theory and providing cleaner proofs of essential facts. To this end, I'll discuss recent progress in boosting a foundational result about norms from equivariant algebra to this more general context.
Location: VH 570 Zoom link: https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327

Ben Spitz, UCLA

Abstract not available

Essential dimension via prismatic cohomology

Jesse Wolfson, UC Irvine

Classical resolvent problems (essential dimension, essential p-dimension, resolvent degree, . . .) ask some form of "How complex is . . . a polynomial, an enumerative problem, a branched cover, a variation of Hodge structure, . . .?" An idea going back to Arnold is that characteristic classes should be able to detect this intrinsic complexity. However, to make this work one must show that the relevant characteristic class remains nonzero under restriction to arbitrary Zariski open subvarieties. In this talk, we describe a new method for solving this restriction problem in many cases using prismatic cohomology. As an application, we prove a conjecture of Brosnan that for a complex abelian variety A, the essential p-dimension of the p-isogeny cover A\to A equals dim A for all but finitely many p. This is joint work with Benson Farb and Mark Kisin.

VH 570 Zoom link: https://umn.zoom.us/j/94317526327?pwd=TkhEU0tRSkljN1l2aXRianBCQzd2QT09 Meeting ID: 943 1752 6327