Cours des Professeurs Permanents de l'IHES

 

The course will focus on rigorous results for the self-avoiding walk model on lattices, with a special emphasis on low-dimensional ones. The model is defined by choosing uniformly at random among random walk paths starting from the origin and without self-intersections. Despite its simple definition, the self-avoiding walk is difficult to comprehend in a mathematically rigorous fashion, and many of the most important problems illustrating standard challenges of critical phenomena remain unsolved. The model is combinatorial in nature but many questions about the stochastic properties of these random paths can be solved by combining nice combinatorial features with probabilistic techniques. In the course, we will describe some of the recent techniques developed in the area, including the use of discrete holomorphicity to understand the model on the hexagonal lattice.

Topological recursion (TR) is a remarkable universal recursive structure that has been found in many enumerative geometry problems, from combinatorics of maps (discrete surfaces), to random matrices, Gromov-Witten invariants, knot polynomials, conformal blocks, integrable systems… An example of topological recursion is the famous Mirzakhani recursion that determines recursively the hyperbolic volumes of moduli spaces. It is a recursion on the Euler characteristic, whence the name "topological" recursion.

A recursion needs an initial data: a "spectral curve" (which we shall define), and the recursion defines the sequence of "TR-invariants" of that spectral curve.

In this series of lectures, we shall:

– define the topological recursion, spectral curves and their TR-invariants, and illustrated with examples.

– state and prove many important properties, in particular how TR-invariants get deformed under deformations of the spectral curve, and how they are related to intersection numbers of moduli spaces of Riemann surfaces, for example the link to Givental formalism.

– introduce the new algebraic approach by Kontsevich-Soibelman, in terms of quantum Airy structures.

– present the relationship of these invariants to integrable systems, tau functions, quantum curves.

– if time permits, we shall present the conjectured relationship to Jones and Homfly polynomials of knots, as an extension of the volume conjecture.

Topological recursion (TR) is a remarkable universal recursive structure that has been found in many enumerative geometry problems, from combinatorics of maps (discrete surfaces), to random matrices, Gromov-Witten invariants, knot polynomials, conformal blocks, integrable systems… An example of topological recursion is the famous Mirzakhani recursion that determines recursively the hyperbolic volumes of moduli spaces. It is a recursion on the Euler characteristic, whence the name "topological" recursion.

A recursion needs an initial data: a "spectral curve" (which we shall define), and the recursion defines the sequence of "TR-invariants" of that spectral curve.

In this series of lectures, we shall:

– define the topological recursion, spectral curves and their TR-invariants, and illustrated with examples.

– state and prove many important properties, in particular how TR-invariants get deformed under deformations of the spectral curve, and how they are related to intersection numbers of moduli spaces of Riemann surfaces, for example the link to Givental formalism.

– introduce the new algebraic approach by Kontsevich-Soibelman, in terms of quantum Airy structures.

– present the relationship of these invariants to integrable systems, tau functions, quantum curves.

– if time permits, we shall present the conjectured relationship to Jones and Homfly polynomials of knots, as an extension of the volume conjecture.

Topological recursion (TR) is a remarkable universal recursive structure that has been found in many enumerative geometry problems, from combinatorics of maps (discrete surfaces), to random matrices, Gromov-Witten invariants, knot polynomials, conformal blocks, integrable systems… An example of topological recursion is the famous Mirzakhani recursion that determines recursively the hyperbolic volumes of moduli spaces. It is a recursion on the Euler characteristic, whence the name "topological" recursion.

A recursion needs an initial data: a "spectral curve" (which we shall define), and the recursion defines the sequence of "TR-invariants" of that spectral curve.

In this series of lectures, we shall:

– define the topological recursion, spectral curves and their TR-invariants, and illustrated with examples.

– state and prove many important properties, in particular how TR-invariants get deformed under deformations of the spectral curve, and how they are related to intersection numbers of moduli spaces of Riemann surfaces, for example the link to Givental formalism.

– introduce the new algebraic approach by Kontsevich-Soibelman, in terms of quantum Airy structures.

– present the relationship of these invariants to integrable systems, tau functions, quantum curves.

– if time permits, we shall present the conjectured relationship to Jones and Homfly polynomials of knots, as an extension of the volume conjecture.

Topological recursion (TR) is a remarkable universal recursive structure that has been found in many enumerative geometry problems, from combinatorics of maps (discrete surfaces), to random matrices, Gromov-Witten invariants, knot polynomials, conformal blocks, integrable systems… An example of topological recursion is the famous Mirzakhani recursion that determines recursively the hyperbolic volumes of moduli spaces. It is a recursion on the Euler characteristic, whence the name "topological" recursion.

A recursion needs an initial data: a "spectral curve" (which we shall define), and the recursion defines the sequence of "TR-invariants" of that spectral curve.

In this series of lectures, we shall:

– define the topological recursion, spectral curves and their TR-invariants, and illustrated with examples.

– state and prove many important properties, in particular how TR-invariants get deformed under deformations of the spectral curve, and how they are related to intersection numbers of moduli spaces of Riemann surfaces, for example the link to Givental formalism.

– introduce the new algebraic approach by Kontsevich-Soibelman, in terms of quantum Airy structures.

– present the relationship of these invariants to integrable systems, tau functions, quantum curves.

– if time permits, we shall present the conjectured relationship to Jones and Homfly polynomials of knots, as an extension of the volume conjecture.

In the `Esquisse d'un programme', Grothendieck proposed studying the action of the absolute Galois group upon the system of profinite fundamental groups of moduli spaces of curves of genus g with n marked points. Around 1990, Ihara, Drinfeld and Deligne independently initiated the study of the unipotent completion of the fundamental group of the projective line with 3 points. It is now known to be motivic by Deligne-Goncharov and generates the category of mixed Tate motives over the integers.  It is closely related to many classical objects such as polylogarithms and multiple zeta values, and has a wide range of applications from number theory to physics.

 

In the first, geometric, half of this lecture series I will explain how to extend this theory to genus one (which generates the theory in all higher genera). The unipotent fundamental groupoid must be replaced with a notion of relative completion, studied by Hain, which defines an extremely rich system of mixed Hodge structures built out of modular forms. It is closely related to Manin's iterated Eichler integrals, the universal mixed elliptic motives of Hain and Matsumoto, and the elliptic polylogarithms of Beilinson and Levin. The question that I wish to confront is whether relative completion stands a chance of generating all mixed modular motives or not. This is equivalent to studying the action of a `motivic' Galois group upon it, and the question of geometrically constructing all generalised Rankin-Selberg extensions.

 

In the second, elementary, half of these lectures, which will be mostly independent from the first, I will explain how the relative completion has a realisation in a new class of non-holomorphic modular forms which correspond in a certain sense to mixed motives. These functions are elementary power series in $q$ and $overline{q}$ and $log |q|$ whose coefficients are periods. They are closely related to the theory of modular graph functions in string theory and also intersect with the theory of mock modular forms.

In the `Esquisse d'un programme', Grothendieck proposed studying the action of the absolute Galois group upon the system of profinite fundamental groups of moduli spaces of curves of genus g with n marked points. Around 1990, Ihara, Drinfeld and Deligne independently initiated the study of the unipotent completion of the fundamental group of the projective line with 3 points. It is now known to be motivic by Deligne-Goncharov and generates the category of mixed Tate motives over the integers.  It is closely related to many classical objects such as polylogarithms and multiple zeta values, and has a wide range of applications from number theory to physics.

 

In the first, geometric, half of this lecture series I will explain how to extend this theory to genus one (which generates the theory in all higher genera). The unipotent fundamental groupoid must be replaced with a notion of relative completion, studied by Hain, which defines an extremely rich system of mixed Hodge structures built out of modular forms. It is closely related to Manin's iterated Eichler integrals, the universal mixed elliptic motives of Hain and Matsumoto, and the elliptic polylogarithms of Beilinson and Levin. The question that I wish to confront is whether relative completion stands a chance of generating all mixed modular motives or not. This is equivalent to studying the action of a `motivic' Galois group upon it, and the question of geometrically constructing all generalised Rankin-Selberg extensions.

 

In the second, elementary, half of these lectures, which will be mostly independent from the first, I will explain how the relative completion has a realisation in a new class of non-holomorphic modular forms which correspond in a certain sense to mixed motives. These functions are elementary power series in $q$ and $overline{q}$ and $log |q|$ whose coefficients are periods. They are closely related to the theory of modular graph functions in string theory and also intersect with the theory of mock modular forms.

 

In the `Esquisse d'un programme', Grothendieck proposed studying the action of the absolute Galois group upon the system of profinite fundamental groups of moduli spaces of curves of genus g with n marked points. Around 1990, Ihara, Drinfeld and Deligne independently initiated the study of the unipotent completion of the fundamental group of the projective line with 3 points. It is now known to be motivic by Deligne-Goncharov and generates the category of mixed Tate motives over the integers.  It is closely related to many classical objects such as polylogarithms and multiple zeta values, and has a wide range of applications from number theory to physics.

 

In the first, geometric, half of this lecture series I will explain how to extend this theory to genus one (which generates the theory in all higher genera). The unipotent fundamental groupoid must be replaced with a notion of relative completion, studied by Hain, which defines an extremely rich system of mixed Hodge structures built out of modular forms. It is closely related to Manin's iterated Eichler integrals, the universal mixed elliptic motives of Hain and Matsumoto, and the elliptic polylogarithms of Beilinson and Levin. The question that I wish to confront is whether relative completion stands a chance of generating all mixed modular motives or not. This is equivalent to studying the action of a `motivic' Galois group upon it, and the question of geometrically constructing all generalised Rankin-Selberg extensions.

 

In the second, elementary, half of these lectures, which will be mostly independent from the first, I will explain how the relative completion has a realisation in a new class of non-holomorphic modular forms which correspond in a certain sense to mixed motives. These functions are elementary power series in $q$ and $overline{q}$ and $log |q|$ whose coefficients are periods. They are closely related to the theory of modular graph functions in string theory and also intersect with the theory of mock modular forms.

 

In the `Esquisse d'un programme', Grothendieck proposed studying the action of the absolute Galois group upon the system of profinite fundamental groups of moduli spaces of curves of genus g with n marked points. Around 1990, Ihara, Drinfeld and Deligne independently initiated the study of the unipotent completion of the fundamental group of the projective line with 3 points. It is now known to be motivic by Deligne-Goncharov and generates the category of mixed Tate motives over the integers.  It is closely related to many classical objects such as polylogarithms and multiple zeta values, and has a wide range of applications from number theory to physics.

 

In the first, geometric, half of this lecture series I will explain how to extend this theory to genus one (which generates the theory in all higher genera). The unipotent fundamental groupoid must be replaced with a notion of relative completion, studied by Hain, which defines an extremely rich system of mixed Hodge structures built out of modular forms. It is closely related to Manin's iterated Eichler integrals, the universal mixed elliptic motives of Hain and Matsumoto, and the elliptic polylogarithms of Beilinson and Levin. The question that I wish to confront is whether relative completion stands a chance of generating all mixed modular motives or not. This is equivalent to studying the action of a `motivic' Galois group upon it, and the question of geometrically constructing all generalised Rankin-Selberg extensions.

 

In the second, elementary, half of these lectures, which will be mostly independent from the first, I will explain how the relative completion has a realisation in a new class of non-holomorphic modular forms which correspond in a certain sense to mixed motives. These functions are elementary power series in $q$ and $overline{q}$ and $log |q|$ whose coefficients are periods. They are closely related to the theory of modular graph functions in string theory and also intersect with the theory of mock modular forms.

 

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.

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.

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.