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.

The gravitational waves detected by LIGO were produced in the final faze of the inward spiraling of two black holes before they collided to produce a more massive black hole. The experiment is entirely consistent with the so called Final State Conjecture of General Relativity according to which generic solutions of the Einstein vacuum equations can be described, asymptotically, by a finite number of Kerr solutions moving away from each other. Though the conjecture is so very easy to formulate and happens to be validated by both astrophysical observations as well as numerical experiments, it is far beyond our current mathematical understanding. In fact even the far simpler and fundamental question of the stability of one Kerr black hole remains wide open.

In my lectures I will address the issue of stability as well as other aspects the mathematical theory of black holes such as rigidity of black holes and the problem of collapse. The rigidity conjecture asserts that all stationary solutions the Einstein vacuum equations must be Kerr black holes while the problem of collapse addresses the issue of how black holes form in the first place from regular initial conditions. Recent advances on all these problems were made possible by a remarkable combination of geometric and analytic techniques which I will try to outline in my lectures.

This crash course will review the theory of the generation of gravitational waves, as well as the theory of the motion and radiation of the premier expected source for gravitational wave interferometric detectors: binary systems.