פעילויות השבוע

In finite dimensional Riemannian geometry, everything behaves nicely — the Riemannian metric induces a distance function, geodesics exist (at least for some time), and so on. In infinite dimensional Riemannian geometry, however, chaos reigns. In this talk I will focus on diffeomorphism groups, and on a particularly important hierarchy of Riemannian metrics on them: right-invariant Sobolev metrics. These arise in many different contexts, from purely mathematical ones, to applications in hydrodynamics and imaging. I will give a brief introduction to these metrics, why we care about them, and what we know (and don‘t know) about their properties. Parts of the talk will be based on joint works with Bob Jerrard and Martin Bauer.

Very little is known about the Tate-Shafarevich groups of abelian varieties. On the one hand, the BSD conjecture predicts that they are finite. On the other hand, heuristics suggest that, for each prime $p$, a positive proportion of elliptic curves $E/\mathbb{Q}$ have $\Sha(E)[p] \ne 0$, and one expects something similar for higher dimensional abelian varieties as well. Yet, despite these expectations, it seems to be an open question whether, for each prime $p$, there exists even a single elliptic curve over $\mathbb{Q}$ with $\Sha(E)[p] \ne 0$. In this talk, I will show that, for each prime $p$, there exists a geometrically simple abelian variety $A/\mathbb{Q}$ with $\Sha(A)[p]\ne 0$. Our examples arise from modular forms with Eisenstein congruences. This is joint work with Ari Shnidman.

A topological dynamical system is said to be Bohr chaotic if for any bounded sequence it possesses a continuous function that correlates with the sequence when evaluated along some orbit. The theme of the lecture will be the relation of this property to an abundance of invariant measures of the system.