Registration is free, but required - please use the linked form.

Zoom links for the lectures will be emailed to participants' registered email addresses.

Lectures will take place weekly on Tuesdays at 4pm Irish Standard Time, with the exception of 16/03/2021, when the talk will begin at 4:30pm GMT.

Each lecture will last between 1-1.5 hours.

Contact: jack(dot)kelly(at)tcd(dot)ie

Description

Let R be a Banach ring. In this lecture series we will explain how derived geometry, in the sense of Toën-Vezzosi, relative to the monoidal model category of simplicial bornological R-modules, provides a universal setting for geometry over R. In particular it contains models for derived algebraic geometry, derived analytic geometry (both Archimedean and non-Archimedean), and in the case that R is the real numbers, smooth geometry. This allows one to construct well-behaved (derived) intersections of analytic spaces, but also, for example, intersections of analytic spaces with algebraic spaces. Moreover the abstract machinery of Toën-Vezzosi's theory means that the (infinity) category of quasi-coherent bornological sheaves on a simplicial bornological (pre)stack comes for free, as do push-forward and pullback functors between such categories. In the analytic case, one can identify a subcategory of sheaves which are closely related to the notion of quasi-coherent sheaves on analytic spaces developed by Ramis-Ruget and Eschmeier-Putinar. We will also give applications, in particular to analytic moduli spaces of Galois representations. Finally, we will introduce some work in progress on bornological spectra, which both provide a model for stable analytic geometry over F_1, and allow us to define bornological generalised cohomology theories.