Activities This Week

A long-standing conjecture of Nicolas Bergeron and Akshay Venkatesh predicts that in closed hyperbolic 3-manifolds, the amount of torsion in the first homology of finite-sheeted normal covers should grow exponentially with the degree of the cover as the covers become larger, at a rate reflecting the volume of the manifold. Yet no finitely presented residually finite group is known to exhibit exponential torsion growth in first homology along an exhausting chain of finite-index normal subgroups.

In this talk I will explain how a two-dimensional lens offers a clearer view of some of the underlying mechanisms that create homological torsion in finite covers, and why obtaining exponential growth may be more tractable in this setting. I will also discuss how these ideas connect to the question of profinite rigidity: how much information about a group is encoded in its finite quotients.

The talk focuses on the general behavior of discrete weakly o-minimal structures, and their definable sets and functions. Comparisons are drawn with the dense weakly o-minimal case and with the discrete o-minimal case. We develop tools to describe definable maps in these structures, and ultimately state a prove a monotonicity theorem. We also state, using the results given so far, how we hope to prove that no definable groups exist in the discrete, weakly o-minimal setting.

In this talk I will give an overview of Popa’s deformation/rigidity theory for von Neumann algebra factors. The main motivation for this theory is the question of when an isomorphism of factors arising from group actions comes from an isomorphism of the groups. After providing some background on early results, examples of using deformation/rigidity to prove structural results will be given. Topics discussed in this talk include deformations, Cartan subalgebras, and intertwining of subalgebras.

There are problems of optimization for which one can optimize locally or one can have a finitely additive measurable outcome, but never both together. This is a connection to the Banach-Tarski Paradox and non-amenable group and semi group action.