La conjecture de conservativité affirme qu’un morphisme entre motifs constructibles est un isomorphisme s’il en est ainsi de l’une des ses réalisations classiques (de Rham, $ell$-adique, etc.). Il s’agit d’une conjecture centrale dans la théorie des motifs ayant des conséquences concrètes sur les cycles algébriques.

Dans ce cours, on s’intéresse à la conjecture de conservativité en caractéristique nulle et, plus précisément, pour la réalisation de de Rham. L’objectif est double :

– D’une part, je parlerai de la tentative de preuve annoncée récemment par l’orateur. L’objectif ici est de décrire suffisamment la structure de l’argument afin d’arriver à l’énoncé problématique et de réaliser l’obstacle qui empêche l’argument d’aboutir.

– D’autre part, je parlerai d’une nouvelle stratégie visant à contourner l’énoncé problématique dans l’argument initial.

La conjecture de conservativité affirme qu’un morphisme entre motifs constructibles est un isomorphisme s’il en est ainsi de l’une des ses réalisations classiques (de Rham, $ell$-adique, etc.). Il s’agit d’une conjecture centrale dans la théorie des motifs ayant des conséquences concrètes sur les cycles algébriques.

Dans ce cours, on s’intéresse à la conjecture de conservativité en caractéristique nulle et, plus précisément, pour la réalisation de de Rham. L’objectif est double :

– D’une part, je parlerai de la tentative de preuve annoncée récemment par l’orateur. L’objectif ici est de décrire suffisamment la structure de l’argument afin d’arriver à l’énoncé problématique et de réaliser l’obstacle qui empêche l’argument d’aboutir.

– D’autre part, je parlerai d’une nouvelle stratégie visant à contourner l’énoncé problématique dans l’argument initial.

Lectures 1-3 are mostly based on our recent work with Linhui Shen.

Given a surface S with punctures and special points on the boundary considered modulo isotopy, and a split semi-simple adjoint group G, we define and quantize moduli spaces Loc(G,S) G-local systems on S, generalising character varieties.

To achieve this, we introduce a new moduli space P(G, S) closely related to Loc(G,S). We prove that it has a cluster Poisson variety structure, equivariant under the action of a discrete group, containing the mapping class group of S. This generalises results of V. Fock and the author, and I. Le.

For any cluster Poisson variety X, we consider the quantum Langlands modular double of the algebra of regular functions on X. If the Planck constant h is either real or unitary, we equip it with a structure of a *-algebra, and construct its principal series of representations.

Combining this, we get principal series representations of the quantum Langlands modular double of the algebras of regular functions on moduli spaces P(G, S) and Loc(G,S).

We discuss applications to representations theory, geometry, and mathematical physics.

In particular, when S has no boundary, we get a local system of infinite dimensional vector spaces over the punctured determinant line bundle on the moduli space M(g,n). Assigning to a complex structure on S the coinvariants of oscillatory representations of W-algebras sitting at the punctures of S, we get another local system on the same spa. We conjecture there exists a natural non-degenerate pairing between these local systems, providing conformal blocks for Liouville / Toda theories.

In Lecture 4 we discuss spectral description of non-commutative local systems on S, providing a non-commutative cluster structure of the latter. It is based on our joint work with Maxim Kontsevich.

Geometry of scalar curvature, that is comparable in scope to symplectic geometry, mediates between two worlds: the domain of rigidity, one sees in convexity and the realm of softness, characteristic of topology, such as the cobordism theory.

The aim of this course is threefold:

1. An overview of old and new  results, mostly, but not exclusively, on the rigidity side, of manifolds X with positive and, more generally, bounded from below scalar curvatures Sc(X), along with a brief introduction to main techniques.

2. Proof of new geometric comparison type inequalities for Riemannian manifolds X with lower bounds on Sc(X) and on mean curvatures of the boundaries of X.

3. Discussion of open problems concerning Sc>0.

Geometry of scalar curvature, that is comparable in scope to symplectic geometry, mediates between two worlds: the domain of rigidity, one sees in convexity and the realm of softness, characteristic of topology, such as the cobordism theory.

The aim of this course is threefold:

1. An overview of old and new  results, mostly, but not exclusively, on the rigidity side, of manifolds X with positive and, more generally, bounded from below scalar curvatures Sc(X), along with a brief introduction to main techniques.

2. Proof of new geometric comparison type inequalities for Riemannian manifolds X with lower bounds on Sc(X) and on mean curvatures of the boundaries of X.

3. Discussion of open problems concerning Sc>0.

Geometry of scalar curvature, that is comparable in scope to symplectic geometry, mediates between two worlds: the domain of rigidity, one sees in convexity and the realm of softness, characteristic of topology, such as the cobordism theory.

The aim of this course is threefold:

1. An overview of old and new  results, mostly, but not exclusively, on the rigidity side, of manifolds X with positive and, more generally, bounded from below scalar curvatures Sc(X), along with a brief introduction to main techniques.

2. Proof of new geometric comparison type inequalities for Riemannian manifolds X with lower bounds on Sc(X) and on mean curvatures of the boundaries of X.

3. Discussion of open problems concerning Sc>0.

Geometry of scalar curvature, that is comparable in scope to symplectic geometry, mediates between two worlds: the domain of rigidity, one sees in convexity and the realm of softness, characteristic of topology, such as the cobordism theory.

The aim of this course is threefold:

1. An overview of old and new  results, mostly, but not exclusively, on the rigidity side, of manifolds X with positive and, more generally, bounded from below scalar curvatures Sc(X), along with a brief introduction to main techniques.

2. Proof of new geometric comparison type inequalities for Riemannian manifolds X with lower bounds on Sc(X) and on mean curvatures of the boundaries of X.

3. Discussion of open problems concerning Sc>0.

Les principales questions abordées dans cette série de cours concernent l’existence locale et globale en temps, explosion en temps fini et la résolution en solitons des solutions de l’équation des ondes non linéaire énergie critique.
Les lectures ne demanderont pas de pré-requis..

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 ­

Lectures 1-3 are mostly based on our recent work with Linhui Shen.

Given a surface S with punctures and special points on the boundary considered modulo isotopy, and a split semi-simple adjoint group G, we define and quantize moduli spaces Loc(G,S) G-local systems on S, generalising character varieties.

To achieve this, we introduce a new moduli space P(G, S) closely related to Loc(G,S). We prove that it has a cluster Poisson variety structure, equivariant under the action of a discrete group, containing the mapping class group of S. This generalises results of V. Fock and the author, and I. Le.

For any cluster Poisson variety X, we consider the quantum Langlands modular double of the algebra of regular functions on X. If the Planck constant h is either real or unitary, we equip it with a structure of a *-algebra, and construct its principal series of representations.

Combining this, we get principal series representations of the quantum Langlands modular double of the algebras of regular functions on moduli spaces P(G, S) and Loc(G,S).

We discuss applications to representations theory, geometry, and mathematical physics.

In particular, when S has no boundary, we get a local system of infinite dimensional vector spaces over the punctured determinant line bundle on the moduli space M(g,n). Assigning to a complex structure on S the coinvariants of oscillatory representations of W-algebras sitting at the punctures of S, we get another local system on the same spa. We conjecture there exists a natural non-degenerate pairing between these local systems, providing conformal blocks for Liouville / Toda theories.

In Lecture 4 we discuss spectral description of non-commutative local systems on S, providing a non-commutative cluster structure of the latter. It is based on our joint work with Maxim Kontsevich.

Lectures 1-3 are mostly based on our recent work with Linhui Shen.

Given a surface S with punctures and special points on the boundary considered modulo isotopy, and a split semi-simple adjoint group G, we define and quantize moduli spaces Loc(G,S) G-local systems on S, generalising character varieties.

To achieve this, we introduce a new moduli space P(G, S) closely related to Loc(G,S). We prove that it has a cluster Poisson variety structure, equivariant under the action of a discrete group, containing the mapping class group of S. This generalises results of V. Fock and the author, and I. Le.

For any cluster Poisson variety X, we consider the quantum Langlands modular double of the algebra of regular functions on X. If the Planck constant h is either real or unitary, we equip it with a structure of a *-algebra, and construct its principal series of representations.

Combining this, we get principal series representations of the quantum Langlands modular double of the algebras of regular functions on moduli spaces P(G, S) and Loc(G,S).

We discuss applications to representations theory, geometry, and mathematical physics.

In particular, when S has no boundary, we get a local system of infinite dimensional vector spaces over the punctured determinant line bundle on the moduli space M(g,n). Assigning to a complex structure on S the coinvariants of oscillatory representations of W-algebras sitting at the punctures of S, we get another local system on the same spa. We conjecture there exists a natural non-degenerate pairing between these local systems, providing conformal blocks for Liouville / Toda theories.

In Lecture 4 we discuss spectral description of non-commutative local systems on S, providing a non-commutative cluster structure of the latter. It is based on our joint work with Maxim Kontsevich.

Lectures 1-3 are mostly based on our recent work with Linhui Shen.

Given a surface S with punctures and special points on the boundary considered modulo isotopy, and a split semi-simple adjoint group G, we define and quantize moduli spaces Loc(G,S) G-local systems on S, generalising character varieties.

To achieve this, we introduce a new moduli space P(G, S) closely related to Loc(G,S). We prove that it has a cluster Poisson variety structure, equivariant under the action of a discrete group, containing the mapping class group of S. This generalises results of V. Fock and the author, and I. Le.

For any cluster Poisson variety X, we consider the quantum Langlands modular double of the algebra of regular functions on X. If the Planck constant h is either real or unitary, we equip it with a structure of a *-algebra, and construct its principal series of representations.

Combining this, we get principal series representations of the quantum Langlands modular double of the algebras of regular functions on moduli spaces P(G, S) and Loc(G,S).

We discuss applications to representations theory, geometry, and mathematical physics.

In particular, when S has no boundary, we get a local system of infinite dimensional vector spaces over the punctured determinant line bundle on the moduli space M(g,n). Assigning to a complex structure on S the coinvariants of oscillatory representations of W-algebras sitting at the punctures of S, we get another local system on the same spa. We conjecture there exists a natural non-degenerate pairing between these local systems, providing conformal blocks for Liouville / Toda theories.

In Lecture 4 we discuss spectral description of non-commutative local systems on S, providing a non-commutative cluster structure of the latter. It is based on our joint work with Maxim Kontsevich.