## Symmetries and correspondences |
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June 21-24 2016, School of Mathematical Sciences University of Nottingham

Speakers: Laurent Lafforgue (IHES) and Olivia Caramello (Paris 7)

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Abstract. The lectures will be concerned with the automorphic transfer from a reductive group G over a global field F to GL(d) which Langlands has conjectured to be induced by any d-dimensional representation r of the dual group of G. The first part of the lectures will explain why this automorphic transfer is equivalent to the existence of some adelic Fourier transform on G, defined locally by r, which verifies some global non-additive Poisson formula. Then we will try to propose a direct non conjectural definition of the Fourier transform at each place and to study its properties.

**Olivia Caramello**

*Motivic toposes*

Abstract. We will present a number of notions and results already obtained in the direction of constructing classifying toposes of Weil-type cohomology theories and associated categories of motives. In particular, we will show that a Q-linear category of mixed motives can be naturally constructed as a suitable subcategory of a topos classifying a regular theory attached to any homology functor with coefficients of characteristic 0; this generalizes Nori’s construction. We will also propose a framework based on atomic two-valued toposes and homogeneous models for analysing the properties of Weil-type cohomology theories and their relations. The first part of the course will be devoted to recalling the necessary topos-theoretic preliminaries of both logical and geometric nature. Papers: Motivic toposes, arxiv:math.AG/1507.06271;
Syntactic categories for Nori motives (with L. Barbieri-Viale and L. Lafforgue),
arxiv:math.AG /1506.06113

*Schedule of the lectures*

June 21 Room C5, Physics Building (Building 22)

11:00-12:00 Laurent Lafforgue

12:15-13:15 Laurent Lafforgue

15:00-16:00 Olivia Caramello

16:30-17:30 Olivia Caramello

June 22 Room C4, Physics Building (Building 22)

9:45-10:45 Laurent Lafforgue

11:00-12:00 Laurent Lafforgue

13:30-14:30 Olivia Caramello

14:45-15:45 Olivia Caramello

June 23 Room A17, Pope Building (Building 27)

10:00-11:00 Laurent Lafforgue

11:15-12:15 Laurent Lafforgue

14:00-15:00 Olivia Caramello

15:15-16:15 Olivia Caramello

June 24 Room A17, Pope Building (Building 27)

10:00-11:00 Laurent Lafforgue

11:15-12:15 Laurent Lafforgue

14:00-15:00 Olivia Caramello

15:15-16:15 Olivia Caramello

Directions to, and a map of, the campus are available at
this page.

Scientific queries about the event can be addressed to Ivan Fesenko.