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.  Some of those numbers show up in the formulas "more often" than other ones — think of $0$, $1$ or $\pi=3.1415926\dots$ — and that gives us good reason to distinguish them from the rest by giving them appropriate "unique" names and studying them in greater depth.  Names can be symbolic and short — like $\pi$ or $G$ — but in most cases such numbers are named after scientists: the Euler–Mascheroni constant, the Planck constant, etc.  The special number $2^{1/12}=1.059463\dots$ appears as the frequency ratio of a semitone (the interval between any two adjacent notes) and, at the same time, indicates that the 100% annual interest on a bank account means about 5.9% interest monthly.  In my lecture, I will discuss the conceptual and historical development of the practice of assigning names to numbers, as well as related issues concerning the calculation of numbers and a deeper "understanding" of them.

With musical contributions by the following artists:

• Alexandra Badea (piano)

• Eva-Maria Hekkelman (cello)
• Sun Woo Park (piano)
• David Prinz (piano)
• Liza Schonlau (vocals)

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. This talk is partly based on joint work with Masataka Iwai and Shin-ichi Matsumura.

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.

In the first part of the talk, I will recall the splitting models of Shimura varieties, and explain how they can be used to construct exotic Hecke correspondences. By using a part of the categorical local Langlands correspondence, I will then deduce geometric realizations of the Jacquet-Langlands correspondence.
In the second part of the talk, I will introduce splitting versions of the affine Deligne-Lusztig varieties, and explain how they can be applied to the Tate conjecture for the special fibers of certain Shimura varieties.