Activities This Week

Negative dependence among participants’ outcomes arises naturally in probabilistic models of tournaments and plays an important role in various asymptotic results, including limit theorems, Poisson approximations, and the behavior of extremal scores. In particular, the property of negative orthant dependence has been shown in several works for different tournament models, usually requiring a separate proof in each case. In this work, we present a unified and more general approach by establishing the stronger property of negative association. This stronger notion of dependence allows us to derive limit distributions for order statistics, such as the maximum and the second-highest scores, even though their exact distributions are not available for general tournament sizes. We illustrate our approach using the round-robin tournament model (paired-comparison model in statistics). We also resolve an open problem and prove that the probability of having a unique winner tends to one as the number of players grows.

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.

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.