Lecture 2: A Twistor Formula for the Tree-Level Gravity S-MatrixThis lecture will obtain a compact twistor formula for the full tree-level gravitational S-matrix beyond the self-dual sector.  It uses an extension of the complex geometry of twistor space of the previous lecture.  In the final formula, all integrations are saturated against delta functions yielding a sum of residues.  These depend on the n momenta and polarization vectors of associated gravity linear waves.Supported by the « 2021 Balzan Prize for Gravitation: Physical and Astrophysical Aspects », awarded to Thibault Damour

Atttention : La première Leçon aura lieu à l’Institut Mathématique d’Orsay, Amphi Yoccoz, le 15 mai à 14hRetrouvez toutes ces informations sur le site de la Fondation Mathématique Jacques Hadamard :https://fondation-hadamard.fr/en/articles/2023/01/20/hadamard-lectures-2023/Abstract:Since Weil, mathematicians have understood that there is a deep analogy between the ordinary integers and polynomials in one variable over a finite field, as well as between number fields and the fields of functions on algebraic curves over finite fields. Using this, we can take classical problems in number theory and consider their analogues involving polynomials over finite fields, to which new geometric techniques can be applied that aren’t available in the classical setting. In this course, I will survey recent progress on such problems.Specifically, I will try to highlight how the geometric perspective produces connections to other areas of mathematics, including how the circle method for counting solutions to Diophantine equations can be used to study the topology of moduli spaces of curves in varieties, how geometric approaches to the Cohen-Lenstra heuristics and their generalizations motivate new results of a purely probabilistic nature, and how the analytic theory of automorphic forms over function fields is connected to geometric Langlands theory.

Atttention : La première Leçon aura lieu à l’Institut Mathématique d’Orsay, Amphi Yoccoz, le 15 mai à 14hRetrouvez toutes ces informations sur le site de la Fondation Mathématique Jacques Hadamard :https://fondation-hadamard.fr/en/articles/2023/01/20/hadamard-lectures-2023/Abstract:Since Weil, mathematicians have understood that there is a deep analogy between the ordinary integers and polynomials in one variable over a finite field, as well as between number fields and the fields of functions on algebraic curves over finite fields. Using this, we can take classical problems in number theory and consider their analogues involving polynomials over finite fields, to which new geometric techniques can be applied that aren’t available in the classical setting. In this course, I will survey recent progress on such problems.Specifically, I will try to highlight how the geometric perspective produces connections to other areas of mathematics, including how the circle method for counting solutions to Diophantine equations can be used to study the topology of moduli spaces of curves in varieties, how geometric approaches to the Cohen-Lenstra heuristics and their generalizations motivate new results of a purely probabilistic nature, and how the analytic theory of automorphic forms over function fields is connected to geometric Langlands theory.

Atttention : La première Leçon aura lieu à l’Institut Mathématique d’Orsay, Amphi Yoccoz, le 15 mai à 14hRetrouvez toutes ces informations sur le site de la Fondation Mathématique Jacques Hadamard :https://fondation-hadamard.fr/en/articles/2023/01/20/hadamard-lectures-2023/Abstract:Since Weil, mathematicians have understood that there is a deep analogy between the ordinary integers and polynomials in one variable over a finite field, as well as between number fields and the fields of functions on algebraic curves over finite fields. Using this, we can take classical problems in number theory and consider their analogues involving polynomials over finite fields, to which new geometric techniques can be applied that aren’t available in the classical setting. In this course, I will survey recent progress on such problems.Specifically, I will try to highlight how the geometric perspective produces connections to other areas of mathematics, including how the circle method for counting solutions to Diophantine equations can be used to study the topology of moduli spaces of curves in varieties, how geometric approaches to the Cohen-Lenstra heuristics and their generalizations motivate new results of a purely probabilistic nature, and how the analytic theory of automorphic forms over function fields is connected to geometric Langlands theory.

Atttention : La première Leçon aura lieu à l’Institut Mathématique d’Orsay, Amphi Yoccoz, le 15 mai à 14hRetrouvez toutes ces informations sur le site de la Fondation Mathématique Jacques Hadamard :https://fondation-hadamard.fr/en/articles/2023/01/20/hadamard-lectures-2023/Abstract:Since Weil, mathematicians have understood that there is a deep analogy between the ordinary integers and polynomials in one variable over a finite field, as well as between number fields and the fields of functions on algebraic curves over finite fields. Using this, we can take classical problems in number theory and consider their analogues involving polynomials over finite fields, to which new geometric techniques can be applied that aren’t available in the classical setting. In this course, I will survey recent progress on such problems.Specifically, I will try to highlight how the geometric perspective produces connections to other areas of mathematics, including how the circle method for counting solutions to Diophantine equations can be used to study the topology of moduli spaces of curves in varieties, how geometric approaches to the Cohen-Lenstra heuristics and their generalizations motivate new results of a purely probabilistic nature, and how the analytic theory of automorphic forms over function fields is connected to geometric Langlands theory.

Atttention : La première Leçon aura lieu à l’Institut Mathématique d’Orsay, Amphi Yoccoz, le 15 mai à 14hRetrouvez toutes ces informations sur le site de la Fondation Mathématique Jacques Hadamard :https://fondation-hadamard.fr/en/articles/2023/01/20/hadamard-lectures-2023/Abstract:Since Weil, mathematicians have understood that there is a deep analogy between the ordinary integers and polynomials in one variable over a finite field, as well as between number fields and the fields of functions on algebraic curves over finite fields. Using this, we can take classical problems in number theory and consider their analogues involving polynomials over finite fields, to which new geometric techniques can be applied that aren’t available in the classical setting. In this course, I will survey recent progress on such problems.Specifically, I will try to highlight how the geometric perspective produces connections to other areas of mathematics, including how the circle method for counting solutions to Diophantine equations can be used to study the topology of moduli spaces of curves in varieties, how geometric approaches to the Cohen-Lenstra heuristics and their generalizations motivate new results of a purely probabilistic nature, and how the analytic theory of automorphic forms over function fields is connected to geometric Langlands theory.

Let M be a geometrically finite hyperbolic manifold, that is, a hyperbolic manifold with a fundamental domain consisting of a finitely-sided polyhedron. There exists a unique measure on the unit tangent bundle invariant under the geodesic flow with maximal entropy, and we consider its lift to the frame bundle. In joint work with Pratyush Sarkar and Wenyu Pan, we prove that the frame flow is exponentially mixing with respect to this measure. To establish exponential mixing, we base ourselves on the countable coding of the flow and a version of Dolgopyat’s method, à la Sarkar-Winter and Tsujii-Zhang. To overcome the difficulty of the fractal structure in applying Dolgopyat’s method, we prove a large deviation property for symbolic recurrence to the large subsets.

We study the problem of determining when a compact group can be realized as the group of isometries of a norm on a finite-dimensional real vector space. This problem turns out to be difficult to solve in full generality, but we manage to understand the connected groups that arise as connected components of isometry groups. The classification we obtain is related to transitive actions on spheres (Borel, Montgomery-Samelson) on the one hand and to prehomogeneous spaces (Vinberg, Sato-Kimura) on the other. Joint work with Martin Liebeck, Assaf Naor and Aluna Rizzoli.

I will give new non-commutative (matrix model) membrane solutions, and discuss the issue of infinite-energy extended objects sweeping out in space-time world-volumes of vanishing mean curvature.  Pour être informé des prochains séminaires vous pouvez vous abonner à la liste de diffusion en écrivant un mail à sympa@listes.math.cnrs.fr avec comme sujet: « subscribe seminaire_mathematique PRENOM NOM »(indiquez vos propres prénom et nom) et laissez le corps du message vide. 

We review recent progress in understanding the resurgent properties of integrable field theories in two dimensions. After a brief recap on elementary notions about Borel resummations, we start with a quick historical detour on the study of the large order behaviour of perturbation theory in quantum field theory (QFT) before the advent of resurgence. We then introduce basic notions of resurgence and apply it on three well-known integrable field theories which are UV-free, develop a mass gap in the IR, admit a 1/N expansion, and present the so called renormalon singularities. The interplay between resurgent properties and the 1/N expansion is discussed. The observable of interest is the free energy in the presence of a chemical potential coupled to a conserved charge, which can be computed exactly with thermodynamic Bethe ansatz techniques and/or large N QFT methods. Results at finite N will also be reviewed.

We review recent progress in understanding the resurgent properties of integrable field theories in two dimensions. After a brief recap on elementary notions about Borel resummations, we start with a quick historical detour on the study of the large order behaviour of perturbation theory in quantum field theory (QFT) before the advent of resurgence. We then introduce basic notions of resurgence and apply it on three well-known integrable field theories which are UV-free, develop a mass gap in the IR, admit a 1/N expansion, and present the so called renormalon singularities. The interplay between resurgent properties and the 1/N expansion is discussed. The observable of interest is the free energy in the presence of a chemical potential coupled to a conserved charge, which can be computed exactly with thermodynamic Bethe ansatz techniques and/or large N QFT methods. Results at finite N will also be reviewed.

We review recent progress in understanding the resurgent properties of integrable field theories in two dimensions. After a brief recap on elementary notions about Borel resummations, we start with a quick historical detour on the study of the large order behaviour of perturbation theory in quantum field theory (QFT) before the advent of resurgence. We then introduce basic notions of resurgence and apply it on three well-known integrable field theories which are UV-free, develop a mass gap in the IR, admit a 1/N expansion, and present the so called renormalon singularities. The interplay between resurgent properties and the 1/N expansion is discussed. The observable of interest is the free energy in the presence of a chemical potential coupled to a conserved charge, which can be computed exactly with thermodynamic Bethe ansatz techniques and/or large N QFT methods. Results at finite N will also be reviewed.