Tag - Stable homology

Dan Petersen: Uniform Twisted Homological Stability

3rd November, 2023

Homological stability is now well established as an organizing principle and computational tool in algebraic topology and other areas. In many cases it is of interest to obtain homological stability with twisted coefficients, and the standard choice of such coefficients are the polynomial coefficient systems. All known approaches to homological stability with polynomial coefficients produce a stable range depending on the degree of polynomiality. I will explain a method of obtaining uniform stable ranges for some classes of groups and coefficients of natural interest. This has important consequences in arithmetic statistics, discussed in the number theory seminar on Nov 2.

Craig Westerland: The Stable Homology of the Braid Group with Coefficients Arising from the Hyperelliptic Representation

2nd November, 2023

The braid group B2g+1 has a description in terms of the hyperelliptic mapping class group of a curve X of genus g. This equips it with an action on V = H1(X), and we may produce a wealth of new representations Sλ(V) by applying Schur functors to V. The goal of this talk is to describe the stable (in g) group homology of these representations. Following an idea of Randal-Williams in the setting of the full mapping class group, one may extract these homology groups as Taylor coefficients of the functor given by the stable homology of the space of maps from the universal hyperelliptic curve to a varying target space. We compute that stable homology by way of a scanning argument, much as in Segal’s original computation of the stable homology of configuration spaces. This is joint work with Bergström, Diaconu, and Petersen. Dan will speak afterwards on the application of these results to the conjecture of Andrade-Keating on moments of quadratic L-functions in the function field setting.

Jordan Ellenberg: Stable Homology and the BKPLR Heuristics Over Function Fields

1st November, 2023

A basic question in arithmetic statistics is:  what does the Selmer group of a random abelian variety look like?  This question is governed by the Poonen-Rains heuristics, later generalized by Bhargava-Kane-Lenstra-Poonen-Rains, which predict, for instance, that the mod p Selmer group of an elliptic curve has size p+1 on average.  Results towards these heuristics have been very partial but have nonetheless enabled major progress in studying the distribution of ranks of abelian varieties.