We give a classification of open Klein topological conformal field theories in terms of Calabi-Yau $A_\infty$-categories endowed with an involution. Given an open Klein topological conformal field theory, there is a universal open-closed extension whose closed part is the involutive version of the Hochschild chains associated to the open part.

I will describe a categorical approach to analytic geometry using the theory of quasi-abelian closed symmetric monoidal categories which works both for Archimedean and non-Archimdedean base fields. In particular I will show how the weak G-topologies of (dagger) affinoid subdomains can be characterized by homological method. I will end by briefly saying how to generalize these results for characterizing open embeddings of Stein spaces. This project is a collaboration with Oren Ben-Bassat and Kobi Kremnizer.

We will discuss our approach to global analytic geometry, based on overconvergent power series and functors of functions. We will explain how slight modifications of it allow us to develop a derived version of global analytic geometry. We will finish by discussing applications to the cohomological study of arithmetic varieties.

I will talk in down to earth terms about several main features of this theory.

We describe how p-adic height pairings allow us to find integral points on hyperelliptic curves, in the spirit of Kim's nonabelian Chabauty program. In particular, we discuss how to carry out this ``quadratic Chabauty'' method over quadratic number fields (joint work with Amnon Besser and Steffen Mueller) and present related ideas to find rational points on bielliptic genus 2 curves (joint work with Netan Dogra).