Sujets en physique mathématique
Ref: 3PN3150
Description
Les lois fondamentales de la natures sont géométriques plutôt qu'algébriques. Ce cours présente
quelques concepts clefs de la physique théorique moderne. Son but est de faire comprendre et
assimiler les méthodes géométriques utilisées en physique.
Période(s) du cours
SM11
Prérequis
1SL3000 (Physique Quantique et statistique)
Syllabus
1. Classical Mechanics vs Symplectic Geometry.
2. Gravitation vs Riemannian Geometry.
3. Physical Fields vs Bundles.
4. Geometrical Berry Phase vs Holonomy.
5. Forces and Potentials vs (Co-)homology. De Rham Theory in Electromagnetism.
6. Topological Insulators vs Homotopy.
7. Defects vs Algebraic Topology.
8. Topological quantum field theory vs Categories.
2. Gravitation vs Riemannian Geometry.
3. Physical Fields vs Bundles.
4. Geometrical Berry Phase vs Holonomy.
5. Forces and Potentials vs (Co-)homology. De Rham Theory in Electromagnetism.
6. Topological Insulators vs Homotopy.
7. Defects vs Algebraic Topology.
8. Topological quantum field theory vs Categories.
Composition du cours
Exposés magistraux + questions des étudiants
Lectures guidée dans livre de référence
Problèmes-type résolus en classe
Série de problèmes à faire par soi-même
Ressources
Enseignant: I. Kornev
Bibliographie: M. Nakahara, Geometry, Topology and Physics, Institute of Physics Publishing, Philadelphia, 1990
Bibliographie: M. Nakahara, Geometry, Topology and Physics, Institute of Physics Publishing, Philadelphia, 1990
Résultats de l'apprentissage couverts par le cours
The objective is to cover a broad range of topics at the expense of giving an in-depth treatment to only a small handful of them. We will introduce and discuss: quantum optics and mechanics from a symplectic geometric point of view, Riemannian geometry and general relativity, connections on fi ber bundles and the standard model for particle physics, and applications of algebraic topology and category theory in physics.
Support de cours, bibliographie
[1] Arnold, Mathematical methods of classical mechanics.
[2] Landau and Lifshitz, Mechanics, Vol.I Course of Theoretical Physics.
[3] Manfredo do Carmo, Riemannian Geometry, Boston, 1993.
[4] S. Carroll, Space Time and Geometry: An Introduction to General Relativity (AddisonWelsey, 2003)
[5] M. Nakahara, Geometry, Topology and Physics, Institute of Physics Publishing, Philadelphia, 1990
[6] Zeidler, Quantum Field Theory, Volume 1-3
[7] R. Penrose, The road to reality
[8] Hatcher, Algebraic Topology, availlable online
[9] Frankel, Geometry of Physics
[10] Chaikin and Lubensky, Principles of condensed matter physics.
[11] Mineev, Topologically stable defects and solitons in ordered media.
[12] Turaev, Quantum Invariants of Knots and 3-manifolds
[2] Landau and Lifshitz, Mechanics, Vol.I Course of Theoretical Physics.
[3] Manfredo do Carmo, Riemannian Geometry, Boston, 1993.
[4] S. Carroll, Space Time and Geometry: An Introduction to General Relativity (AddisonWelsey, 2003)
[5] M. Nakahara, Geometry, Topology and Physics, Institute of Physics Publishing, Philadelphia, 1990
[6] Zeidler, Quantum Field Theory, Volume 1-3
[7] R. Penrose, The road to reality
[8] Hatcher, Algebraic Topology, availlable online
[9] Frankel, Geometry of Physics
[10] Chaikin and Lubensky, Principles of condensed matter physics.
[11] Mineev, Topologically stable defects and solitons in ordered media.
[12] Turaev, Quantum Invariants of Knots and 3-manifolds