Grothendieck conference

From tomorrow, Tuesday the 24th of May, until Saturday the 28th of May, there will be, physically at Chapman University and virtually on Zoom, a conference celebrating Grothendieck's work.

The programme, which is available from the conference website, is very rich and consists of several contributions highlighting the relevance of Grothendieck's ideas across many different fields of knowledge.

I will give my own talk, entitled "On the 'unifying notion' of topos", on Friday 27th at 10 am PST (7pm CEST).

The Zoom link to the conference is
Meeting ID: 968 3948 3231

Looking forward to seeing many of you attending the conference!

Grothendieck: la moisson

The podcast of today's broadcast at France Culture "Grothendieck: la moisson" is already available on the Radio website, which also provides several references for the general public to learn more about toposes and Grothendieck's vision.

The associated Twitter account can be accessed here.

Thanks again to Nicolas Martin and all the staff of "La méthode Scientifique" for their invitation! I look forward to join you again in a few weeks, always with Alain Connes and Laurent Lafforgue, for the second part of the broadcast.

An update: a transcription of the emission by Denise Chemla is available here.

An interview for France Culture

On Thursday 17 February, from 16 to 17 French time, I will be a guest, together with Alain Connes and Laurent Lafforgue, at the scientific radio broadcast La Méthode Scientifique of France Culture, to talk about Grothendieck and crucial themes addressed in his recently published text Récoltes et Semailles, such as toposes and ethical issues in mathematics.

Looking forward to it!

Ontologies and toposes

The slides of the double talk given by Laurent Lafforgue and myself at the recent Huawei workshop on semantics are now available:

Our talk is based on an expository article written by AI theorists on the general theme of “ontology” and “knowledge representation”. We explain that the theory of Grothendieck toposes, of their geometric presentations by sites and of their linguistic descriptions in terms of first-order theories provides the means to incarnate in mathematical objects amenable to computations what could have seemed at first sight loose philosophical views. We present a number of important basic ideas about Grothendieck toposes and motivations for beginning to study them, and give an idea of the possibilities of the theory of “toposes as bridges” and its significance for the problem of “knowledge representation”.

Récoltes et Semailles

The famous autobiographical work in two volumes by Alexander Grothendieck has just been officially published by Gallimard!

This book is a must for anyone who wants to learn about the vision inspiring Grothendieck's discoveries and reflect on ethical and sociological issues in mathematics and beyond.

I have been invited by the Editor to contribute a short text presenting the book, to be included in a booklet sold together with it and also containing texts by other mathematicians. Here is it:

An update: Sayantan Roy @sayantan-roy has kindly translated my text into English. Here is his translation:

Lectures Grothendieckiennes

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.

Relative topos theory via stacks

I am glad to announce the first version of our joint work with Riccardo Zanfa on relative toposes:

Relative topos theory via stacks

Here is the abstract:

We 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 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. Another key ingredient in our theory is a notion of relative site, which allows one to represent arbitrary geometric morphisms towards a fixed base topos of sheaves on a site as structure morphisms induced by relative sites over that site.

We shall progressively release expanded versions of this text contaning new developments in the directions sketched in the introduction.

This work has been recently presented at the conference Toposes online:

The following video focuses on the specialization of the fundamental adjunction in the setting of presheaves (or discrete fibrations):

On the "unifying notion" of topos

The video of my recent talk "Toposes as unifying spaces: historical aspects and prospects" at the Workshop in honor of Alexander Grothendieck’s legacy at the Universidad Nacional de Colombia is now available on YouTube:

This talk (whose slides can be downloaded here) discusses how the unifying concept of topos was introduced and conceived by Grothendieck, as well as the future prospects provided by the theory of toposes as ‘bridges’.

It is a shorter (and partially different) version of the lecture on the same subject I gave in 2018 at the ENS for the series Lectures Grothendieckiennes organized by Frédéric Jaëck (whose slides are available here):

In fact, I have recently finished writing (in French) my contribution for the Proceedings volume of that lecture series, which will be published by Spartacus: this paper can be downloaded here. An English translation is also in preparation.