Given a symplectic variety X, we construct a canonical category, which should be thought of as a category of modules over a quantization of the loop space of X. For example, if X = T^*M is a cotangent bundle, the category in question is a variant of the category of D-modules on the loop space of M. We discuss the potential application to the relative Langlands program, sheaves of vertex algebras (chiral quantizations), and line defects in quantum field theory. Based on work in progress with D. Kubrak.

Physics has long supplied mathematics with some of its central equations. I will argue that Transformers, the engine of large language models, are becoming a source of the same kind. Starting from raw code, under various scaling assumptions, one can interpret Transformers as an interacting particle system in continuous time; tools from PDE and kinetic theory then explain empirically observed phenomena such as representation collapse and “lost in the middle”. Unlike in physics, however, the laws here are designed rather than given, so mathematical understanding can be written back into the architecture. I will close with some thoughts on what this two-way exchange means for applied mathematics.

In the last two decades a remarkable surge of results in dispersive PDE have addressed qualitative features of the dynamics of large solutions, based on methods such as the Kenig-Merle approach. On the other hand, effective quantitative results appear to be almost unavailable. In this talk I will present recent work with Jaime Gomez, giving an effective quantitative estimate of scattering solutions near the scattering threshold for the cubic nonlinear Klein-Gordon equation, as described by Nakanishi-Schlag. The main ingredient is an induction on energy type argument, but in the context of a focussing nonlinear wave equation.