L6

We define a version of algebraic K-theory for bornological algebras, using the recently developed continuous K-theory by Efimov. In the commutative setting, we prove that this invariant satisfies descent for various topologies that arise in analytic geometry, generalising the results of Thomason-Trobaugh for schemes. Finally, we prove a version of the Grothendieck-Riemann-Roch Theorem for analytic spaces. Joint work with Jack Kelly and Federico Bambozzi. 

Let G be a reductive group over a finite field F of characteristic p. I will present work with Tzu-Jan Li in which we determine the endomorphism algebra of the Gelfand-Graev representation of the finite group G(F) where the coefficients are taken to be l-adic integers, for l a good prime of G distinct from p. Our result can be viewed as a finite-field analogue of the local Langlands correspondence in families. 

Let X --> S be a family of algebraic varieties parametrized by an infinitesimal disk S, possibly of mixed characteristic. The Bloch conductor conjecture expresses the difference of the Euler characteristics of the special and generic fibers in algebraic and arithmetic terms. I'll describe a proof of some new cases of this conjecture, including the case of isolated singularities. The latter was a conjecture of Deligne generalizing Milnor's formula on vanishing cycles. 

This is joint work with Massimo Pippi; our methods use derived and non-commutative algebraic geometry. 

Plethysms lie at the intersection of representation theory and algebraic combinatorics. We give a recursive formula for a family of plethysm coefficients encompassing those involved in Foulkes' Conjecture. We also describe some applications, such as to the stability of plethysm coefficients and Sylow branching coefficients for symmetric groups. This is joint work with Y. Okitani.

The Aldous-Lyons conjecture from probability theory states that every (unimodular) infinite graph can be (Benjamini-Schramm) approximated by finite graphs. This conjecture is an analogue of other influential conjectures in mathematics concerning how well certain infinite objects can be approximated by finite ones; examples include Connes' embedding problem (CEP) in functional analysis and the soficity problem of Gromov-Weiss in group theory. These became major open problems in their respective fields, as many other long-standing open problems, that seem unrelated to any approximation property, were shown to be true for the class of finitely-approximated objects. For example, Gottschalk's conjecture and Kaplansky's direct finiteness conjecture are known to be true for sofic groups, but are still wide open for general groups.

In 2019, Ji, Natarajan, Vidick, Wright and Yuen resolved CEP in the negative. Quite remarkably, their result is deduced from complexity theory, and specifically from undecidability in certain quantum interactive proof systems. Inspired by their work, we suggest a novel interactive proof system which is related to the Aldous-Lyons conjecture in the following way: If the Aldous-Lyons conjecture was true, then every language in this interactive proof system is decidable. A key concept we introduce for this purpose is that of a Subgroup Test, which is our analogue of a Non-local Game. By providing a reduction from the Halting Problem to this new proof system, we refute the Aldous-Lyons conjecture.

This talk is based on joint work with Lewis Bowen, Alex Lubotzky, and Thomas Vidick.

No special background in probability theory or complexity theory will be assumed.

I will describe some recent developments in random walks on Gromov-hyperbolic spaces. I will focus in particular on the notions of Schottky sets and pivoting technique introduced respectively by Boulanger-Mathieu-S-Sisto and Gouëzel and mention some consequences. The talk will be introductory; I will not assume specialized knowledge in probability theory.

During the 14th to the 16th centuries CE, a succession of Indian scholars, collectively referred to as the Kerala school, made remarkable contributions in the fields of mathematics and astronomy. Mādhava of Saṅgamagrāma, a gifted mathematician and astronomer, is considered the founder of this school, and is perhaps best known for discovering an infinite series for pi, among other achievements. Subsequently, Mādhava's lineage of disciples, consisting of illustrious names such as Parameśvara, Dāmodara, Nīlakaṇṭha, Jyeṣṭhadeva, Śaṅkara Vāriyar, Citrabhānu, Acyuta Piṣaraṭi etc., made numerous important contributions of their own in the fields of mathematics and astronomy. Later scholars of the Kerala school flourished up to the 19th century. This talk will provide a historical overview of the Kerala school and highlight its important contributions.

If \(F\) is a free group and \(F/N\) is a presentation of a group \(G\), there is a natural way to turn the abelianisation of \(N\) into a \(\mathbb ZG\)-module, known as the relation module of the presentation. The images of normal generators for \(N\) yield \(\mathbb ZG\)-module generators of the relation module, but 'lifting' \(\mathbb ZG\)-generators to normal generators cannot always be done by a result of Dunwoody. Nevertheless, it is an open problem, known as the relation gap problem, whether the relation module can have strictly fewer \(\mathbb ZG\)-module generators than \(N\) can have normal generators when \(G\) is finitely presented. In this talk I will survey what is known and what is not known about this problem and its variations and discuss some recent progress for groups with a cyclic relation module.

I'll talk about the Morse local-to-global property and try to convince you that is a good property. There are three reasons. Firstly, it is satisfied by many examples of interest. Secondly, it allows to prove many theorems. Thirdly, it sits nicely in the larger program of classifying groups up to quasi-isometry and it has connections with open questions.

This talk will serve as an introduction to the outer automorphism group of a free group, its properties and the objects used to study it: especially train track maps (with various adjectives) and Culler--Vogtmann outer space. If time allows I will discuss recent work joint with Hillen, Lyman and Pfaff on stretch factors in rank 3, but the goal of the talk will be to introduce the topic well rather than to speedrun towards the theorem.