Author Archives: Olivia Caramello

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

On women and mathematics

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On the occasion of the release in France of the movie Marguerite’s theorem, I was interviewed at Radio France with Mélanie Guenais and Ariane Mézard about women and mathematics.

The discussion, lasting for about one hour, has touched several different themes, from statistics related to the numbers of female students and researchers in mathematics to ways to improve mathematics education at large.

You may listen to the podcast here.

A generalisation of Diaconescu’s theorem for relative toposes

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A new paper, written jointly with my Ph.D. student Léo Bartoli, is available from the ArXiv:

On morphisms of relative toposes

The paper carries out a systematic investigation of the functors between sites which induce morphisms between relative toposes.

It culminates in a generalisation of Diaconescu’s theorem for relative toposes, formulated in the language of fibrations and stacks according to the foundations for relative topos theory introduced in the paper Relative topos theory via stacks.

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.

Visions in mathematics: from Grothendieck to the present day

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The Grothendieck Institute is hosting a conference for the general public on 14 June 2023 at the Institut Henri Poincaré in Paris. This meeting, entitled “Visions in mathematics: from Grothendieck to the present day“, will explore, with talks by Mateo Carmona, Alain Connes, Stéphane Dugowson, Johanna Grothendieck, Laurent Lafforgue and myself, the issues raised by the person and visionary work of A. Grothendieck from a philosophical, literary and sociological perspective. It will also discuss mathematical themes such as topos theory and its future in today’s mathematical landscape, industrial research and the dissemination of scientific knowledge.

The conference is free of charge and open to all upon prior registration on the Institute’s website.

Looking forward to seeing many of you there!

Talk at Logica Universalis Webinar

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On Wednesday the 17th of May at 4 CET I will give a talk at the Logica Universalis Webinar on the contents of my paper “The unification of Mathematics via Topos Theory”, originally written in 2010 and recently published in the book “Logic in Question” (Studies in Universal Logic, Springer-Birkhäuser, 2022).

The talk will be accompanied by a short presentation of the Grothendieck Institute by Laurent Lafforgue.

Looking forward to seeing many of you there!