I will talk about a result on meromorphic continuation of Euler products over primes p of definable p-adic or motivic integrals, and applications to zeta functions of groups. If time permitting, I'll state an analogue for counting rational points of bounded height in some adelic homogeneous spac

 This talk will discuss the so-called ``generic cohomology’’ of function fields over algebraically closed fields, from the point of view of motives and/or Zariski geometry. In particular, I will describe some interesting connections between cup products, algebraic dependence, and (geometric) valuation theory. As an application, I will mention a new result which reconstructs higher-dimensional function fields from their generic cohomology, endowed with some additional motivic data. 

   Everyone welcome!
 

Given a Shimura variety, I will show how to define a corresponding two-sorted structure. Based on work of Chris Daw and Adam Harris, we will study what is needed for the class of this structures to be categorical. Of course, an introduction to Shimura varieties will be given.
 

In this talk, we will explain how the counting theorems of Pila and Wilkie lead to a conditional proof of the aforementioned conjecture. In particular, we will explain how to generalise the work of Habegger and Pila on a product of modular curves. 
Habegger and Pila were able to prove that the Zilber-Pink conjecture holds in such a product if the so-called weak complex Ax and large Galois orbits conjectures are true. In fact, around the same time, Pila and Tsimerman proved a stronger statement than the weak complex Ax conjecture, namely, the Ax-Schanuel conjecture for the $j$-function. We will formulate Ax-Schanuel and large Galois orbits conjectures for general Shimura varieties and attempt to imitate the Habegger-Pila strategy. However, we will encounter an additional difficulty in bounding the height of a pre-special subvariety.

This is joint work with Jinbo Ren.