The volume of proceedings of the lecture series Lectures Grothendieckiennes given in the academic year 2017-2018 at the Ecole Normale Superieure (Paris) is now available!
This book, which celebrates Grothendieck's mathematical heritage, is prefaced by Peter Scholze and features contributions by Pierre Cartier, Olivia Caramello, Alain Connes, Laurent Lafforgue, Colin McLarty, Gilles Pisier, Jean-Jacques Szczeciniarz and Fernando Zalamea.
My contribution is available here:
The videos of the lectures are available from YouTube at this link.
The videos of the talks at the recent school "Unifying themes in Geometry" are now available on YouTube, with links from the conference website. From the website you will also be able to download the slides used during the presentations.
Enjoy the talks!
Tomorrow (Monday 22 November 2021, at 9:40 Central European Time) I will give a talk on "Deductive systems and Grothendieck topologies" for the Dagstuhl Seminar Geometric Logic, Constructivisation, and Automated Theorem Proving.
The abstract is as follows:
I will show that the classical proof system of geometric logic over a given geometric theory is equivalent to new proof systems based on the notion of Grothendieck topology. These equivalences result from a proof-theoretic interpretation of the duality between the quotients of a given geometric theory and the subtoposes of its classifying topos. Interestingly, these alternative proof systems turn out to be computationally better-behaved than the classical one for many purposes, as I will illustrate by discussing a few selected applications.
To attend the talk, you may click here.
Next Tuesday I shall give a talk for ItaCa Fest 2021 on my work in progress with Riccardo Zanfa providing new foundations for relative topos theory based on stacks:
Relative topos theory via stacksIn this talk, based on joint work with Riccardo Zanfa, we shall introduce new foundations for relative topos theory based on stacks. One of the central results in our theory is an adjunction between the category of (relatively small) toposes over the topos of sheaves on a given site (C, J) and that of C-indexed categories. This represents a wide generalization of the classical adjunction between presheaves on a topological space and bundles over it, and allows one to interpret several constructions on sheaves and stacks in a geometrical way; in particular, it leads to fibrational descriptions of direct and inverse images of sheaves and stacks, as well as to a geometric understanding of the sheafification process. It also naturally allows one to regard any Grothendieck topos as a ‘petit’ topos associated with a ‘gros’ topos, thereby providing an answer to a problem posed by Grothendieck in the seventies.
14:30 (Italian time)ItaCa Fest 2021
Here is the Zoom link to attend the talk: https://zoom.us/j/94880770089?pwd=clgxK2VkVEE5Ymw5ME1QWktiWExUZz09
Thanks to the ItaCa Fest organizers for this invitation, and looking forward to seeing many of you there!
Next week I shall give two (video-recorded) talks, one in Italian and another one in English, on unification and morphogenesis from a topos-theoretic perspective:
Topoi come 'ponti’ unificanti: una morfogenesi matematicaIn questo seminario affronterò il tema dell’unificazione in matematica, e della dualità tra unità e molteplicità, dal punto di vista della teoria dei ‘ponti’ topos-teoretici. Discuterò in particolare il senso in cui il tipo di unificazione realizzato da questa teoria rappresenta una vera e propria ‘morfogenesi matematica’, intesa come insieme di dinamiche di differenziazione a partire da un’unità.
Slides available here.
Unification and morphogenesis: a topos-theoretic perspectiveWe shall present some philosophical principles underlying the theory of toposes as unifying `bridges' in mathematics. More specifically, after reviewing the various types of unification which occur in mathematics, we shall discuss the way in which the connections established by topos-theoretic ‘bridges’ give rise to an authentic mathematical morphogenesis.
Slides available here.
16:00 (French time) Seminario di Logica e Filosofia della Scienza,
University of Palermo
Many thanks to the organizers of these events, Proff. Pierluigi Graziani and Gianluigi Oliveri.