The goal of this course is to give a construction of contact homology in the sense of Eliashberg–Givental–Hofer. I will begin with an introduction to contact geometry, pseudo-holomorphic curves as introduced by Gromov, and some target applications of contact homology. The main focus will then be on extracting enough enumerative information (i.e. « virtual fundamental cycles ») out of the relevant moduli spaces of pseudo-holomorphic curves. I will present a general framework for doing this, and then discuss the specific application to contact homology.

The goal of this course is to give a construction of contact homology in the sense of Eliashberg–Givental–Hofer. I will begin with an introduction to contact geometry, pseudo-holomorphic curves as introduced by Gromov, and some target applications of contact homology. The main focus will then be on extracting enough enumerative information (i.e. « virtual fundamental cycles ») out of the relevant moduli spaces of pseudo-holomorphic curves. I will present a general framework for doing this, and then discuss the specific application to contact homology.

The goal of this course is to give a construction of contact homology in the sense of Eliashberg–Givental–Hofer. I will begin with an introduction to contact geometry, pseudo-holomorphic curves as introduced by Gromov, and some target applications of contact homology. The main focus will then be on extracting enough enumerative information (i.e. « virtual fundamental cycles ») out of the relevant moduli spaces of pseudo-holomorphic curves. I will present a general framework for doing this, and then discuss the specific application to contact homology.

Suite des cours donnés en octobre et novembre 2015

A la fin des années 90, les liens entre transport optimal, entropie et courbure de Ricci étaient mis au jour (Jordan-Kinderlehrer-Otto, Otto-Villani); quelques années plus tard, ce liens étaient exploités pour démarrer l'étude systématique du "point de vue synthétique" de la courbure de Ricci (Lott-Sturm-Villani), un domaine en progression constante depuis lors. La résolution récente de plusieurs questions ouvertes majeures suggère que le moment est venu de faire un bilan; c'est l'objectif de ce cours. On y trouvera notamment une nouvelle preuve du théorème d'isopérimétrie de Lévy-Gromov (Cavalletti-Mondino).

Suite des cours donnés en octobre et novembre 2015

A la fin des années 90, les liens entre transport optimal, entropie et courbure de Ricci étaient mis au jour (Jordan-Kinderlehrer-Otto, Otto-Villani); quelques années plus tard, ce liens étaient exploités pour démarrer l'étude systématique du "point de vue synthétique" de la courbure de Ricci (Lott-Sturm-Villani), un domaine en progression constante depuis lors. La résolution récente de plusieurs questions ouvertes majeures suggère que le moment est venu de faire un bilan; c'est l'objectif de ce cours. On y trouvera notamment une nouvelle preuve du théorème d'isopérimétrie de Lévy-Gromov (Cavalletti-Mondino).

Lecture 1 : A brief introduction to noncommutative geometry with emphasis on the essential tools used in physics.
 
Lecture 2 : Classification of finite spaces and basis for geometric unification.
 
Lecture 3 : Spectral action and Standard Model of Particle Physics.
 
Lecture 4 : Order one condition and physics beyond Standard Model.

Seiberg-Witten theory maps supersymmetric four-dimensional gauge theories with extended supersymmetry to algebraic completely integrable systems. For large class of such integrable systems the phase space is the moduli space of solutions of self-dual hyperKahler equations and their low-dimensional descendants. In particular, the list of such integrable systems includes Hitchin systems defined on Riemann surfaces with singularities at marked points (two-dimensional PDE), monopoles on circle bundles over surfaces (three-dimensional PDE or circle-valued Hitchin system) and instantons on torically fibered hyperKahler manifolds (four-dimensional PDE or elliptically valued Hitchin system). Deformations of four-dimensional gauge theory by curved backgrounds correspond to the quantization of the associated algebraic integrable systems. Quantization of Hitchin systems has relation to geometric Langlands correspondence and to the Toda two-dimensional conformal theory with Wg-algebra symmetry. Quantization of g-monopole and g-instanton moduli spaces relates to the representation theory of Drinfeld-Jimbo quantum affine algebras (and their rational and elliptic versions, Yangians and elliptic groups), associated respectively to g in the monopole case (circle-valued Hitchin) and to the central extension of the loop algebra of g in the instanton case (elliptically valued Hitchin). It is expected that there exists an analogue of geometric Langlands correspondence for quantization of the monopole and instanton algebraic integrable system (circle-valued and elliptically-valued Hitchin).

Seiberg-Witten theory maps supersymmetric four-dimensional gauge theories with extended supersymmetry to algebraic completely integrable systems. For large class of such integrable systems the phase space is the moduli space of solutions of self-dual hyperKahler equations and their low-dimensional descendants. In particular, the list of such integrable systems includes Hitchin systems defined on Riemann surfaces with singularities at marked points (two-dimensional PDE), monopoles on circle bundles over surfaces (three-dimensional PDE or circle-valued Hitchin system) and instantons on torically fibered hyperKahler manifolds (four-dimensional PDE or elliptically valued Hitchin system). Deformations of four-dimensional gauge theory by curved backgrounds correspond to the quantization of the associated algebraic integrable systems. Quantization of Hitchin systems has relation to geometric Langlands correspondence and to the Toda two-dimensional conformal theory with Wg-algebra symmetry. Quantization of g-monopole and g-instanton moduli spaces relates to the representation theory of Drinfeld-Jimbo quantum affine algebras (and their rational and elliptic versions, Yangians and elliptic groups), associated respectively to g in the monopole case (circle-valued Hitchin) and to the central extension of the loop algebra of g in the instanton case (elliptically valued Hitchin). It is expected that there exists an analogue of geometric Langlands correspondence for quantization of the monopole and instanton algebraic integrable system (circle-valued and elliptically-valued Hitchin).