Mathematical Colloquium

Geometric representation theory uses geometric tools to construct and analyse algebraic structures like the representations of matrix groups. One key tool is the theory of $D$-modules, which are modules over the differential operators on a given space. It is not surprising that this framework is particularly powerful over the complex numbers, where the analysis of differential equations is reasonably well understood. In this colloquium talk, I will describe an analogous theory over the field of $p$-adic numbers, develop (large parts of) a six-functor formalism for holonomic analytic $D$-modules, and mention a couple of representation-theoretic applications.

Twenty years of research in cluster theory have established deep links between cluster algebras, surface geometry and representation theory. In this talk, I will show how cluster structures can be defined using surface geometry, with curves corresponding to cluster variables or to rigid objects in associated categories. The classical Ptolemy relations give rise to exchange phenomena or mutation in the associated cluster structures.

The first part of the talk will review results and open questions in the arithmetic theory of elliptic curves, related to the Birch and Swinnerton-Dyer conjecture. I will then describe the use of $p$-adic methods on the subject, concerned with the study of $p$-adic $L$-functions via techniques arising from $p$-adic Hodge theory.

Brownian and Langevin dynamics have been the mainstays of computational science for well over a century. Thermostat methods like Nosé dynamics originated in the 1980s as simplified models to mimic properties of thermal baths in molecular simulation. In this talk I will address the unification of these different approaches, their discretization, and their properties as general purpose tools for large scale computation in the physical sciences, machine learning and statistics.

TBA (Here is the abstract of the talk.....)

TBA (Here is the abstract of the talk.....)