Activities This Week

Coarse structures are used to study the large scale geometry of a space. For example, although the integers and the real line are different topologically, they look the same from “far away”, in the sense that any geometric configuration in the real line can be approximated by one in the integers, up to some uniformly bounded error. A coarse group is a group object in the category of coarse spaces, for example, this means the group operation is only “coarsely associative,” etc. In joint work with Federico Vigolo we study coarse groups. This talk will be an advertisement for our work, as we walk through examples that illustrate some of our main results, and connections to other subjects.

Recently, Ofek, Pandey, and Shalit have defined a version of Banach-Mazur distances on the space of isomorphism classes of finite-dimensional complete Pick spaces. By the universality theorem of Agler and McCarthy, every finite-dimensional complete Pick space on n points is equivalent to a subspace of the Drury-Arveson space spanned by n kernels at points of the unit ball of some C^d. We propose to study the space of projections on finite-dimensional multiplier coinvariant subspaces of the Drury-Arveson space. The metric on this space is induced by the norm. We show that if we restrict ourselves to the subspace of projection on spaces spanned by distinct n kernels, then this space is homeomorphic to the symmetrized polyball. It then follows that the invariant distance obtained induces the same topology on the space of isomorphism classes of complete Pick space as the Banach-Mazur distance of Ofek, Pandey, and Shalit. Time permuting we will show a potential application of this idea

For a smooth projective hyperbolic curve Y/Q the set of rational points Y(Q) is finite by Faltings’ Theorem. Grothendieck’s section conjecture predicts that this set can be described via Galois sections of the étale fundamental group of Y. On the other hand, the non-abelian Chabauty method produces p-adic analytic functions which conjecturally cut out Y(Q) as a subset of Y(Qp). We relate the two conjectures and discuss the example of the thrice-punctured line, where non-abelian Chabauty is used to prove a local-to-glocal principle for the section conjecture.