Contact homology and virtual fundamental cycles (1/5)
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.
Contact homology and virtual fundamental cycles (2/5)
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.
Contact homology and virtual fundamental cycles (3/5)
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.
Contact homology and virtual fundamental cycles (4/5)
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.
Contact homology and virtual fundamental cycles (5/5)
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.
La théorie synthétique de la courbure de Ricci (1/2)
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).
La théorie synthétique de la courbure de Ricci (2/2)
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).
Spectral Geometric Unification (4/4)
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.
