I will start with an aspect of mathematics that is well understood that is the Mendelian dynamics in the spaces of alleles. (This is described in Mendelian Dynamics and Sturtevant’s Paradigm in the « recent » section on my website.)
Also I touch upon in this context on the categorical view on the entropy in dynamics as in In a Search for a Structure, Part 1: On Entropy, also in the « recent » section).
(2-3 lectures)

Then I will elaborate on the Poincaré-Sturtevant idea of describing geometries of spaces X by samples of probability measures on the set subsets of X, where Poincaré had in mind the reconstruction of the Euclidean geometry by the Brain and Sturtevant used it to make a genomic map of a chromosome of drosophila.
(1 lecture)

Also I dedicate a lecture to mathematical problems related to the structure and functions of proteins.
I conclude by speculations on further possible mathematical « unfoldings » of messages conveyed by molecular ­

Universal mixed elliptic motives are certain local systems over a modular curve that are endowed with additional structure, such as that of a variation of mixed Hodge structure. They form a tannakian category. The coordinate ring of its fundamental group is a Hopf algebra in a category of mixed Tate motives.

This course will be an introduction to universal mixed elliptic motives, which were defined with Makoto Matsumoto, and a report on more recent developments. One focus will be on the structure of the tannakian fundamental group of the category of mixed elliptic motives over M1,1. In particular, we will explain that it is an extension of GL2 by a prounipotent group whose Lie algebra is generated by Eisenstein series and has non-trivial relations coming from cusp forms. We will also discuss the relation of mixed elliptic motives to mixed Tate motives via specialization to the Tate curve and the nodal cubic.

Universal mixed elliptic motives are certain local systems over a modular curve that are endowed with additional structure, such as that of a variation of mixed Hodge structure. They form a tannakian category. The coordinate ring of its fundamental group is a Hopf algebra in a category of mixed Tate motives.

This course will be an introduction to universal mixed elliptic motives, which were defined with Makoto Matsumoto, and a report on more recent developments. One focus will be on the structure of the tannakian fundamental group of the category of mixed elliptic motives over M1,1. In particular, we will explain that it is an extension of GL2 by a prounipotent group whose Lie algebra is generated by Eisenstein series and has non-trivial relations coming from cusp forms. We will also discuss the relation of mixed elliptic motives to mixed Tate motives via specialization to the Tate curve and the nodal cubic.

Universal mixed elliptic motives are certain local systems over a modular curve that are endowed with additional structure, such as that of a variation of mixed Hodge structure. They form a tannakian category. The coordinate ring of its fundamental group is a Hopf algebra in a category of mixed Tate motives.

This course will be an introduction to universal mixed elliptic motives, which were defined with Makoto Matsumoto, and a report on more recent developments. One focus will be on the structure of the tannakian fundamental group of the category of mixed elliptic motives over M1,1. In particular, we will explain that it is an extension of GL2 by a prounipotent group whose Lie algebra is generated by Eisenstein series and has non-trivial relations coming from cusp forms. We will also discuss the relation of mixed elliptic motives to mixed Tate motives via specialization to the Tate curve and the nodal cubic.

Universal mixed elliptic motives are certain local systems over a modular curve that are endowed with additional structure, such as that of a variation of mixed Hodge structure. They form a tannakian category. The coordinate ring of its fundamental group is a Hopf algebra in a category of mixed Tate motives.

This course will be an introduction to universal mixed elliptic motives, which were defined with Makoto Matsumoto, and a report on more recent developments. One focus will be on the structure of the tannakian fundamental group of the category of mixed elliptic motives over M1,1. In particular, we will explain that it is an extension of GL2 by a prounipotent group whose Lie algebra is generated by Eisenstein series and has non-trivial relations coming from cusp forms. We will also discuss the relation of mixed elliptic motives to mixed Tate motives via specialization to the Tate curve and the nodal cubic.

The aim of the course is to describe the Riemann-Hilbert correspondence for holonomic D-modules in the irregular case and its applications to integral transforms with irregular kernels, especially the Laplace transform. The course will start with a detailed exposition of the theory of indsheaves (including sheaves on the subanalytic site) and its applications to the indsheaf of holomorphic temperate functions.

Télécharger le résumé détaillé du cours.

Consulter la page web du cours.

The aim of the course is to describe the Riemann-Hilbert correspondence for holonomic D-modules in the irregular case and its applications to integral transforms with irregular kernels, especially the Laplace transform. The course will start with a detailed exposition of the theory of indsheaves (including sheaves on the subanalytic site) and its applications to the indsheaf of holomorphic temperate functions.

Télécharger le résumé détaillé du cours.

Consulter la page web du cours.

The aim of the course is to describe the Riemann-Hilbert correspondence for holonomic D-modules in the irregular case and its applications to integral transforms with irregular kernels, especially the Laplace transform. The course will start with a detailed exposition of the theory of indsheaves (including sheaves on the subanalytic site) and its applications to the indsheaf of holomorphic temperate functions.

Télécharger le résumé détaillé du cours.

Consulter la page web du cours.

The aim of the course is to describe the Riemann-Hilbert correspondence for holonomic D-modules in the irregular case and its applications to integral transforms with irregular kernels, especially the Laplace transform. The course will start with a detailed exposition of the theory of indsheaves (including sheaves on the subanalytic site) and its applications to the indsheaf of holomorphic temperate functions.

Télécharger le résumé détaillé du cours.

Consulter la page web du cours.

The aim of the course is to describe the Riemann-Hilbert correspondence for holonomic D-modules in the irregular case and its applications to integral transforms with irregular kernels, especially the Laplace transform. The course will start with a detailed exposition of the theory of indsheaves (including sheaves on the subanalytic site) and its applications to the indsheaf of holomorphic temperate functions.

Télécharger le résumé détaillé du cours.

Consulter la page web du cours.

The aim of the course is to describe the Riemann-Hilbert correspondence for holonomic D-modules in the irregular case and its applications to integral transforms with irregular kernels, especially the Laplace transform. The course will start with a detailed exposition of the theory of indsheaves (including sheaves on the subanalytic site) and its applications to the indsheaf of holomorphic temperate functions.

Télécharger le résumé détaillé du cours.

Consulter la page web du cours.

Singular support is an invariant that can be attached to a coherent sheaf

on a derived scheme which is quasi-smooth (a.k.a. derived locally complete

intersection). This invariant measures how far a given coherent sheaf is

from being perfect. We will explain how the subtle difference between

"coherent" and "perfect" is responsible for the appearance of  Arthur 

parameters in the context of geometric Langlands correspondence.

Télécharger le résumé du cours.

Consulter la page web du cours.