Activities This Week

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 various results, some old and some new, on the structure of the groups of points of an abelian variety over a finite field. The talk will focus on the case of varieties of large dimension over a fixed finite field. In this regime, the Weil bounds allow for the possibility of the exponent of the group staying bounded as the dimension grows. I will explain that at least in the case of Jacobians this cannot be the case. Part of the talk is based on recent joint work with Daniel Keliher.

Kac’s lemma is a classical result in ergodic theory. It asserts that the expected number of

iterates that it takes a point from a measurable set A to return to the set A under an ergodic

probability-preserving transformation is equal to the inverse of the measure of A. As we will discuss in this seminar, there is a natural generalization of Kac’s lemma that applies to probability preserving actions of an arbitrary countable group (and beyond). As an application, we will show that that any ergodic action of a countable group admits a countable generator. The content of this work is based on a joint article with Benjamin Weiss

https://doi.org/10.48550/arXiv.2410.18488

Yotam will give a basic introduction to Nilpotent orbits in Lie algebras. Eitan will use this to introduce the Whittaker models in the generality explained in the paper: generalized Whittaker models.