Talk

In this talk I will present a Hasse principle for divisibility of rational points in algebraic groups, introduced by Dvornicich and Zannier in 2001, and motivated by a particular case of the Hasse principle on quadratic forms and by the Grunwald-Wang Theorem. I will give an overview on what is known so far and what answers are still missing. I will also link this question to other interesting problems, for example on torsion points on abelian varieties.

This is based on joint works with Laura Paladino, Rocco Chirivì, and Nirvana Coppola.

Mathematical facts are often represented through expressions (also known as formulas) that feature numerous numbers.

We show that there are closed, aspherical, smooth 4-manifolds that are homeomorphic but not diffeomorphic. This is joint work with Davis, Hayden, Ruberman, and Sunukjian.

In this talk we present, for any integers $0\leq \nu \leq n$, a set of inequalities satisfied by the Chern classes of any minimal complex projective variety of dimension $n$ and numerical dimension $\nu$. In the cases where $\nu$ is either very small or very large compared with $n$, this recovers many previously known results, notably of Miyaoka and others. We demonstrate that these inequalities are sharp by providing an explicit characterisation of those varieties achieving the equality; our proof, in particular, resolves the Abundance conjecture in this situation.

I will explain how to construct exotic Hecke correspondences between the special fibers of different PEL type Shimura varieties, at possibly ramified primes. These can be used to construct new geometric realizations of the Jacquet-Langlands correspondence, as well as verify generic instances of the Tate conjecture for the special fibers of these Shimura varieties, generalizing the work of Xiao-Zhu in the unramified case. The key is to resolve the integral models by the splitting models of Pappas-Rapoport.