U of T number theory seminar (2026–27)

The local Langlands correspondence for a reductive group \(G\) over \(\mathbb{Q}_p\) posits a correspondence between certain smooth representations \(\pi\) of \(G(\mathbb{Q}_p)\) and certain \(G\)-valued (in an appropriate sense) representations \(\varphi\) of \(\operatorname{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)\). While brought into clearer perspective by recent work, the precise nature of what is meant by \(\varphi\) corresponding to \(\pi\) has historically been somewhat vague but, there is a natural guess: the characters of \(\varphi\) and \(\pi\) should match. This does not quite make sense as traces for \(\varphi\) take as inputs Galois-group elements \(\tau\), and traces on \(\pi\) take as inputs certain functions \(f\) on \(G(\mathbb{Q}_p)\). Thus, it is natural to guess that to each \(\tau\) there is an \(f_\tau\) and one should require that \(\operatorname{tr}(\tau \mid \varphi) = \operatorname{tr}(f_\tau \mid \pi)\). While such a naive statement cannot be true, it is a conjecture of Scholze--Shin that an appropriate refinement does hold. The question then becomes: what are the functions \(f_\tau\)? In this talk I will try to roughly explain the definition of these functions and their relationship to Shimura varieties, the so-called Langlands--Kottwitz--Scholze method.

In this talk, I will discuss a dynamical analogue of a classical number-theoretic question concerning bounds on the greatest common divisors of two integer sequences. I will present some recent progress on this problem and highlight several open directions for future research. Finally, I will explain how this question relates to the Dynamical Mordell--Lang Conjecture, a central topic in algebraic and arithmetic dynamics.