Domoschool 2022

Next week I will be in Domodossola giving a research course on topos theory for the 2022 edition of the International Alpine School of Mathematics and Physics.

The title of my course is Grothendieck toposes, invariants and ‘bridges’. The course will be an introduction to the theory of Grothendieck toposes, with a specific emphasis on the invariants that one can define on them. The point of view that we shall take is the one provided by the theory of toposes as ‘bridges’, which we shall present and illustrate through a selection of notable examples. We shall also discuss the application of these techniques to the investigation and discovery of dualities, equivalences and correspondences in mathematics and beyond.

Videos of the Grothendieck Conference

The videos of the conference in honour of Grothendieck held in May at Chapman University are now available from Youtube.

My own talk at the conference was entitled "The unifying 'notion' of topos":

The slides can be downloaded here. A written text of my presentation will be available soon.

Talk at inter-disciplinary conference of the theme of "Construction"

Tomorrow, I will give at Timeworld Paris 2022, an interdisciplinary conference organized by Laurence Honnorat's company Innovaxiom, a popularizing talk on my research work entitled "Comment construire des 'ponts' en mathématiques?".

A full program of the event is available at the conference website.

I hope to see many of you there!

UPDATE: The slides of my talk are available here.

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  
https://chapman.zoom.us/j/96839483231?from=addon
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.

A course on toposes as 'bridges'

In the next weeks, starting from this Friday at 14:00 London time, Laurent Lafforgue will give an online course on toposes as 'bridges' at the University of Warwick.

The programme is as follows:

Lecture I: Grothendieck topologies, sheaves, toposes and points.

Date: 18 February 2022 (Friday) at 14:00-15:00 (London time).

Abstract: The purpose of this first lecture will be to introduce the notion of Grothendieck topology on a category, the associated notions of sheaves and toposes, and the derived notion of point of a topos. A special attention will be given to the problem of generation of Grothendieck topologies which, as will become clear in the third lecture, is of great significance and magnitude. Comparing the representations of objects of an arbitrary topos in terms of a presenting site and their evaluations at points provides some extremely general form of non-linear Fourier decompositions and Fourier transforms.


Lecture II: Linguistic descriptions of points and first-order geometric theories.

Date: 25 February 2022 (Friday) at 14:00-15:00 (London time).

Abstract: It will be shown how any presentation of a Grothendieck topos by a site allows to give a linguistic description of set-based points and generalised points of a topos. These descriptions make up bridges from geometry to languages, i.e. to words and grammar rules. They also provide a good way to introduce the general notions of first-order languages and first-order geometric theories.


Lecture III: Classifying toposes, toposes as bridges and the equivalence between first-order provability and generation of Grothendieck topologies.

Date: 4 March 2022 (Friday) at 14:00-15:00 (London time).

Abstract: It will be shown that any geometric first-order theory defines a "functor of models" which associates to any topos the category of the models of this theory with coefficients in this topos, and that this functor of models is always representable by a so-called "classifying topos": it is characterized by the property that models of the theory identify with points of this associated classifying topos. The theory of classifying toposes, which was developped by W. Lawvere and the school of categorical logic in the 1970's, building on some seminal ideas of Grothendieck, was given new impetus with the technique of "toposes as bridges" introduced by O. Caramello in her 2009 PhD thesis. This technique consists in exploiting the fact that any topos can be represented by a double infinite diversity of presenting sites and of geometric theories, in order to develop a general theory of relations between mathematical theories. A particular example of that is the interpretation in terms of quotient theories of the invariant of toposes consisting in their ordered sets of subtoposes. It provides an equivalence between the general problem of provability in the context of first-order geometric theories and the problem of generation of Grothendieck topologies on small categories.

For more information, including the links to attend the lectures, please visit the course webpage.

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”.

Working group on proofs and Grothendieck topologies

A working group of about 20 researchers has formed to investigate computational aspects of the methodology 'toposes as bridges', with particular reference to the proof-theoretic equivalences established in Chapters 3 and 8 of my book Theories, Sites, Toposes: Relating and studying mathematical theories through topos-theoretic 'bridges' and described in these slides.

Bridge between Grothendieck topologies and quotients

The organizer is Laurent Lafforgue, and the meetings will take place at the Huawei Lagrange Center for Mathematics and Computation in Paris, starting from the first one, which has happened today.

A final goal is to implement these techniques on a computer, to automatically generate mathematical results by exploiting the capacity of 'bridges' to significantly transform the level of complexity of notions and results. Back in 2010, when I first evoked this possibility in the paper The unification of Mathematics via Topos Theory, that idea was regarded with a lot of skepticism, as something almost too good to be true. Now, the time is ripe to start making that dream into reality.