16 | IHES Annual Report 2020

Laurent LAFFORGUE Mathematics, permanent professor since 2000,

Huawei Chair of Algebraic Geometry.

PUBLICATIONS

Le principe de fonctorialité de Langlands comme un problème de généralisation de la loi d'addition Pre-publication accepted (108 pages) in the Asterisk collection (pending review).

Symétries, invariants et quotients Notes of a lecture given at the Università degli studi dell'Insubria (402 pages); book in preparation.

With O. Caramello Sites et topos de Grothendieck : une introduction Book in preparation (600 pages).

DISTINCTIONS

Prix Peccot, Collège de France (1996) CNRS Bronze Medal (1998) Clay Research Award (2000) Prix Jacques Herbrand, Académie des sciences de Paris (2001) Fields Medal (2002) Doctor Honoris Causa, Notre Dame University, USA (2011) Académie des sciences de Paris, member

Editor of: Annales des sciences mathématiques du Québec Moscow Mathematical Journal (MMJ) Afrika Matematika Milan Journal of Mathematics

CONFERENCES

China (PR.) The 3rd World Laureates Forum, Shanghai (30 October - 1st November) The Way of Mathematics (video presentation)

France Huawei Young Talents Program, Friends of IHES, IHES, Bures-sur-Yvette (6 November) Le pouvoir créateur des catégories : Elements d'Histoire et nouvelles perspectives (video presentation)

Franco-Chinese Institute of Chemistry Beijing, BUCT-Paris Curie Engineering School (16 November) Quelques distinctions fondamentales des mathématiques (video presentation)

Italy SISSA 2020, Scuola Internazionale Superiore di Studi Avanzati, Trieste (16 December) The creative power of categories: elements of History and some new perspectives (video presentation)

This year, LAURENT LAFFORGUE focused mainly on supporting the development of Grothendieck's topos theory and its applications, and aimed at creating one or several schools. He was a privileged interlocutor of Olivia Caramello, particularly on her work on the subject of relative toposes and their presentation in terms of parametrized sites or parametrized theories , a theme that goes beyond the framework for first-order theories, for the theory of classifying toposes established in the 1970s. In addition, he taught aspects of topos theory to two ENS students, one on the theme of cohomological duality within the framework of toposes, i.e. the six

operations formalism , the other related to his lectures: algebraic topology, functional analysis (focusing on the recent work of Clausen-Scholze), logic (focusing on the equivalence established by Olivia Caramello between the rules of inference of intuitionist logic and the axioms of Grothendieck's topologies), and programming (focusing on the current work of Bennequin-Belfiore applying toposes to the study of deep neural networks).

SCIENTIFIC ACTIVITY