פעילויות השבוע

I will give an overview of some recent and some older results on rational dynamics and on rational difference equations. The main model theoretic tool here is the theory ACFA, the model companion of the theory of difference fields.

Locally analytic representations of p-adic Lie groups with Q_p coefficients are powerful tools in p-adic Hodge theory and the p-adic Langlands program. This perspective reveals important differential structures, such as the Sen and Casimir operators. A few years ago, Rodrigues Jacinto and Rodriguez Camargo developed a ”solid“ version of this theory using the language of condensed mathematics, which provides more robust homological tools (comparison theorems, spectral sequences…) for studying these representations. This talk will present work that extends this solid theory to a much broader class of mixed characteristic coefficients, such as F_p((X)) or Z_p[[X]]<p/x>, as well as semilinear representations. I will conclude by exploring how these ideas relate to mixed characteristic phenomena in p-adic Hodge theory and the Langlands program.

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.