Category Archives: Talks

Course on “Relative toposes for artificial general intelligence”

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In the forthcoming weeks, in the framework of my D’Alembert Chair at the Université Paris-Saclay, I shall give a course at the MICS Laboratory of CentraleSupélec on “Relative toposes for artificial general intelligence“.

ABSTRACT:

Topos theory can be described as the science of invariants. The theory of relative toposes consists in methods and techniques for studying toposes in relation to each other. Viewing toposes as spaces embodying information, this theory notably paves the way for the development of new, very dynamical and structural forms of modelling, both of ‘real’ entities and phenomena, and of (natural or artificial) learning processes. In particular, it naturally leads to the design of systems implementing principles of meta-learning (in the sense of learning taking place at different levels of abstraction constructed on top of each other).

The first part of the course will provide a conceptual introduction to the theory of relative toposes and its relevance for AI, while the second will present some first applications of this theory in connection with the modelling and solution of Raven progressive matrices and, more generally, of ARC-type problems.

N.C. No previous knowledge of topos theory is necessary for understanding the key ideas and methods presented in the course.

DATES:

– Thursday 26 June 2025, 14-16, Amphi SC.046 (Peugeot), Bouygues building (9 Rue Joliot Curie, 91190 Gif-sur-Yvette).
– Thursday 3 July 2025, 14-16, Amphi SC.046 (Peugeot), Bouygues building (9 Rue Joliot Curie, 91190 Gif-sur-Yvette).

Looking forward to seeing many of you there!

Relative toposes and meta-learning

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I am glad to share the slides of my recent talk on Relative toposes and meta-learning at the Computing and AI Summit in London.

In this presentation I give a conceptual introduction to the theory of relative toposes and discuss its relevance for modelling learning processes which build on top of existing knowledge through a sequence of steps lying at increasing levels of abstraction.

The ultimate aim of artificial meta-learning should be to mimic the distinctive, multi-layered way in which human learning unfolds, whilst leveraging the superior processing capabilities of machines.

Relative toposes and meta-learningDownload

Talk at Deducteam – ENS Paris-Saclay

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On Friday the 3rd of May, at 10am, I will give a talk on “Deductive systems and Grothendieck topologies” at the seminar of the research group Deducteam of the École normale supérieure Paris-Saclay.

In the talk 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. Remarkably, these new systems, which are related to the classsical one through topos-theoretic ‘bridges’, are computationally much better-behaved for many purposes, as I will illustrate by discussing a few notable applications.

We are looking forward to a computer implementation of these techniques allowing to exploit all the computational advantages deriving from the multiplication of points of view which is made possible by topos theory.

You can attend the talk in presence (in Room 1Z71 of the ENS Paris-Saclay) or online, at this Zoom link.

More information is available at the seminar page.

Looking forward to seeing many of you there!

Course on toposes at the University of Paris-Saclay

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In the coming weeks, starting from Thursday the 14th of March, I will give an introductory course on topos theory at the Laboratoire MICS of CentraleSupélec (Université Paris-Saclay).

More details, including the content and schedule of the lectures, can be found at the course webpage.

Everybody is welcome!

Visit to Lyon

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On Tuesday the 5th of March I will give a talk at the ENS de Lyon, in the framework of their Seminar “Logique, Mathématiques, Informatique, Raisonnement“, about the work on relative topos theory that I have been pursuing in the last years, starting with the paper Relative topos theory via stacks (with R. Zanfa), and which I am currently developing with my Ph.D. students at the Grothendieck Institute.

The seminar will take place at 18 in the Amphi A / Grande Nef 1-2 of the Campus Monod of the ENS de Lyon; you may download the poster here.

Looking forward to it!

Syntactic learning

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The slides of my recent talk at the AI workshop at RL China are available below.

In this presentation I propose an interpretation of learning processes in terms of the notions of mathematical theory and proof, and advocate for the importance of empowering artificial learnings systems with large formal vocabularies that will serve for expressing the concepts (and relations between them) that they will learn from data.

The aim is to obtain more robust and structured forms of learning with generalisation capabilities, and greater resilience and adaptability, mimicking the distinctive features of human intelligence.

We also consider this project as an essential step for arriving at a toposic theory of semantic information; indeed, syntax and semantics are interwined (think, for instance, of the syntactic construction of classifying toposes).

I look forward to experimentally testing these ideas with our team at the Lagrange Center in Paris.

Syntactic Learning Via Topos TheoryDownload

Slides: “Toposic Fraïssé-Galois theory and motivic toposes”

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I am glad to share the slides of my recent talk at the ICBS Satellite Conference on Algebraic and Arithmetic Geometry, where I explained how Fraïssé theory and Galois theory can be extended and linked through a unified topos-theoretic framework that can possibly be applied to the problem of constructing motives and understanding the “independence of l” properties of l-adic cohomology.

TalkICBSDownload

Videos of the Paris event of the Grothendieck Institute

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The videos of the recent conference organized by the Grothendieck Institute at the Institut Henri Poincaré are now available from the Institute’s YouTube channel:

The event consisted in a round table with Alain Connes, Johanna Grothendieck and Laurent Lafforgue, moderated by Stéphane Dugowson and preceded by two brief presentations by Mateo Carmona, Coordinator of the Centre for Grothendieckian Studies, and myself.

Enjoy the videos!

Visit to China

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From the 10th to the 26th of July I will be visiting the BIMSA, a world-class research institution established by the Chinese Ministry of Science and Technology and led by Prof. Shing-Tung Yau, to participate in the Inaugural International Congress of Basic Science (ICBS).

I will also give a plenary talk on the 12th at the ICBS Satellite Conference on Algebraic and Arithmetic Geometry on Toposic Galois-Fraïssé theory and motivic toposes; the abstract is as follows:

We will present an abstract topos-theoretic framework for building Galois-type theories in a variety of different mathematical  contexts : this unifies and generalizes Grothendieck’s theory of  “Galois categories” and Fraïssé’s construction of homogeneous structures in model theory. This theory notably allows one to construct fundamental groups in many classical contexts such as finite groups, finite graphs, motives and many more. We will in particular present an approach based on it for building “motivic toposes” and investigating the independence from l of l-adic cohomology.

Slides: “Relative topos theory via stacks: an introduction”

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The slides of my introductory course to relative toposes recently given at the Lagrange Center are now available:

RelativeToposTheoryViaStacksAnIntroductionDownload