Ergodic Theory is a powerful tool in the study of linear groups. When trying to crystallize its role, emerges the theory of AREAs, that is Algebraic Representations of Ergodic Actions, which provides a categorical framework for various previously studied concepts and methods. Roughly, this theory extends the focus of Representation Theory from Groups to Group Actions, exploiting the tension between Ergodic Theory and Algebraic Geometry. In this series of talks I will introduce this theory and survey some of its applications, focusing on Superrigidity and Arithmeticity results.

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IHES Covid-19 regulations:

– all the participants who will attend the event in person will have to keep their mask on in indoor spaces
and where the social distancing is not possible;
– speakers will be free to wear their mask or not at the moment of their talk if they feel more comfortable
without it;
– Up to 25 persons in the conference room, every participant will be asked to be able to provide a health pass
– Over 25 persons in the conference room, every participant will be asked to provide a health pass which will
be checked at the entrance of the conference room.

==================================================================

Ergodic Theory is a powerful tool in the study of linear groups. When trying to crystallize its role, emerges the theory of AREAs, that is Algebraic Representations of Ergodic Actions, which provides a categorical framework for various previously studied concepts and methods. Roughly, this theory extends the focus of Representation Theory from Groups to Group Actions, exploiting the tension between Ergodic Theory and Algebraic Geometry. In this series of talks I will introduce this theory and survey some of its applications, focusing on Superrigidity and Arithmeticity results.

==================================================================

IHES Covid-19 regulations:

– all the participants who will attend the event in person will have to keep their mask on in indoor spaces
and where the social distancing is not possible;
– speakers will be free to wear their mask or not at the moment of their talk if they feel more comfortable
without it;
– Up to 25 persons in the conference room, every participant will be asked to be able to provide a health pass
– Over 25 persons in the conference room, every participant will be asked to provide a health pass which will
be checked at the entrance of the conference room.

==================================================================

Ergodic Theory is a powerful tool in the study of linear groups. When trying to crystallize its role, emerges the theory of AREAs, that is Algebraic Representations of Ergodic Actions, which provides a categorical framework for various previously studied concepts and methods. Roughly, this theory extends the focus of Representation Theory from Groups to Group Actions, exploiting the tension between Ergodic Theory and Algebraic Geometry. In this series of talks I will introduce this theory and survey some of its applications, focusing on Superrigidity and Arithmeticity results.

==================================================================

IHES Covid-19 regulations:

– all the participants who will attend the event in person will have to keep their mask on in indoor spaces
and where the social distancing is not possible;
– speakers will be free to wear their mask or not at the moment of their talk if they feel more comfortable
without it;
– Up to 25 persons in the conference room, every participant will be asked to be able to provide a health pass
– Over 25 persons in the conference room, every participant will be asked to provide a health pass which will
be checked at the entrance of the conference room.

==================================================================

Liouville conformal field theory (LCFT hereafter), introduced by Polyakov in his 1981 seminal work "Quantum geometry of bosonic strings", can be seen as a random version of the theory of Riemann surfaces. LCFT appears in Polyakov's work as a 2d version of the Feynman path integral with an exponential interaction term. Since then, LCFT has emerged in a wide variety of contexts in the physics literature and in particular recently in relation with 4d supersymmetric gauge theories (via the AGT conjecture).

 

A major issue in theoretical physics was to solve the theory, namely compute the correlation functions. In this direction, an intriguing formula for the three point correlations of LCFT was proposed in the middle of the 90's by Dorn-Otto and Zamolodchikov-Zamolodchikov, the celebrated DOZZ formula.

 

The purpose of the course is twofold (based on joint works with F. David, A. Kupiainen and R. Rhodes). First, I will present a rigorous probabilistic construction of Polyakov's path integral formulation of LCFT. The construction is based on the Gaussian Free Field. Second, I will show that the three point correlation functions of the probabilistic construction indeed satisfy the DOZZ formula. This establishes an explicit link between probability theory (or statistical physics) and the so-called conformal bootstrap approach of LCFT.

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.

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.

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.

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.