Differential Geometry and Topology in Physics, Spring 2023

Syllabus

Lecture notes 

lecture1 (Euler characteristics, supersymmetric quantum mechanics, manifolds)

lecture2 (tangent spaces, vector fields, tangent bundles, orientation)

lecture3 (cotangent bundles, differential forms)

lecture4 (Stokes theorem, de Rham cohomology, metric) 

lecture5 (Harmonic forms, Hodge theorem, Poincare duality, Riemannian geometry)

lecture6 (Riemannian geometry, Gauss-Bonnet theorem, Einstein equations)

lecture7 (Symplectic geometry, Hamiltonian system, Arnold-Liouville theorem)

lecture8 (Calogero-Moser system)

lecture9 (simplicial homology groups, homotopy)

lecture10 (cohomology groups, Lefschetz fixed-point theorem, Poincare-Hopf theorem)

lecture11 (Abelian Chern-Simons theory, fundamental groups)

lecture12 (Lie groups, Lie algebras)

lecture13 (Spherical harmonics, Vector bundles)

lecture14 (principal G-bundles, connections, curvatures, Yang-Mills action)

lecture15 (characteristic classes, Chern-Weil theory)

I have taught the same course in 2017, 2019, and 2021. But I have updated the lecture note by making corrections and adding more content.

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