Title: The duality between spaces and categories from the derived perspective
Abstract: The correspondence between spaces on the one side, and algebraic structures on the other side, is a very general mathematical phenomenon. The purpose of this series of lecture is to revisit this correspondence from the modern point of view of derived algebraic geometry and to present some of its recent developments.
In this first lecture, I will mainly focus on some of the historical aspects of this correspondence, starting from the seminal works of Tannaka and Gelfand, concerning the reconstruction of a compact topological group in terms of its linear representations, and the reconstruction of a compact space in terms of its ring of continuous functions. I will explain how Grothendieck reinterpreted and pursued these original ideas using the language of categories, and how this has led him to a new vision: a striking correspondance between spaces and categories. This notably drove him to his famous theory of motives, and to envision his program on infinity-categories and infinity-stacks as a general context to express general duality statements. In the final part of the talk, I will briefly mention some modern incarnations of the duality spaces categories: the first one concerning moduli of sheaves and enumerative geometry, and a second one concerning algebraic models of homotopy types. These two examples, as well as others, will be discussed and explained in more detail in the following lectures of the series.
Lecture 1: 07 July 2026, 1600 to 1700
Lecture 2: 08 July 2026, 1200 to 1300
Lecture 3: 09 July 2026, 1200 to 1300
Lecture 4: 10 July 2026, 0930 to 1030
About the speaker: Bertrand Toen’s main research interests concern “derived algebraic geometry”, a domain at the interface of algebraic geometry and algebraic topology. He has obtained his PhD from the University of Toulouse in 1999, under the supervision of Carlos Simpson and Joseph Tapia, and is currently a permanent researcher at the Institut Mathématiques de Toulouse. He has been invited to speak at the ICM (2014), and was awarded the Sophie Germain prize (2019) and the silver medal from CNRS (2023).
This lecture series is part of the Discussion Meeting "Enumerative Geometry and Categorification"