Differential Geometry and Topology in Physics, Spring 2021

Syllabus

Lecture notes and Videos

lecture1 (Euler characteristics, supersymmetric quantum mechanics, manifolds)

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

lecture3 (cotangent bundles, differential forms, Stokes theorem)

lecture4 (de Rham cohomology, metric, harmonic forms, Hodge theorem, Poincare duality)

lecture5 (Riemannian geometry, Gauss-Bonnet theorem, Einstein equations)

lecture6 (Symplectic geometry, Hamiltonian system, Arnold-Liouville theorem)

lecture7 (simplicial homology groups, homotopy)

lecture8 (cohomology groups, Lefschetz fixed-point theorem, Poincare-Hopf theorem)

lecture9 (fundamental groups, homotopy groups)

lecture10 (Lie groups, Lie algebras, vector bundles)

lecture11 (principal G-bundles, connections, curvatures, Yang-Mills action)

lecture12 (characteristic classes, Chern-Weil theory)

lecture13 (index theorem, Hirzebruch-Riemann-Roch theorem, anomaly, supersymmetry)

lecture14 (moduli space of flat connections, Chern-Simons theory, TQFT axiom)

lecture15 (Moduli spaces)

I have taught the same course in 2017 and 2019. 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.