Differential Geometry and Topology in Physics, Spring 2023

Lecture 1 Euler characteristics, supersymmetric quantum mechanics, manifolds

Lecture 2 Tangent spaces, vector fields, tangent bundles, orientation

Lecture 3 Cotangent bundles, differential forms

Lecture 4 Stokes theorem, de Rham cohomology, metric

Lecture 5 Harmonic forms, Hodge theorem, Poincaré duality, Riemannian geometry

Lecture 6 Riemannian geometry, Gauss–Bonnet theorem, Einstein equations

Lecture 7 Symplectic geometry, Hamiltonian system, Arnold–Liouville theorem

Lecture 8 Calogero–Moser system

Lecture 9 Simplicial homology groups, homotopy

Lecture 10 Cohomology groups, Lefschetz fixed-point theorem, Poincaré–Hopf theorem

Lecture 11 Abelian Chern–Simons theory, fundamental groups

Lecture 12 Lie groups, Lie algebras

Lecture 13 Spherical harmonics, vector bundles

Lecture 14 Principal \(G\)-bundles, connections, curvatures, Yang–Mills action

Lecture 15 Characteristic classes, Chern–Weil theory

Lecture notes are written by referring to various sources without mentioning them. Comments are welcome.

Syllabus