Differential Geometry and Topology in Physics, Spring 2021

Lecture 1 Euler characteristics, supersymmetric quantum mechanics, manifolds

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

Lecture 3 Cotangent bundles, differential forms, Stokes theorem

Lecture 4 de Rham cohomology, metric, harmonic forms, Hodge theorem, Poincaré duality

Lecture 5 Riemannian geometry, Gauss–Bonnet theorem, Einstein equations

Lecture 6 Symplectic geometry, Hamiltonian system, Arnold–Liouville theorem

Lecture 7 Simplicial homology groups, homotopy

Lecture 8 Cohomology groups, Lefschetz fixed-point theorem, Poincaré–Hopf theorem

Lecture 9 Fundamental groups, homotopy groups

Lecture 10 Lie groups, Lie algebras, vector bundles

Lecture 11 Principal G-bundles, connections, curvatures, Yang–Mills action

Lecture 12 Characteristic classes, Chern–Weil theory

Lecture 13 Index theorem, Hirzebruch–Riemann–Roch theorem, anomaly, supersymmetry

Lecture 14 Moduli space of flat connections, Chern–Simons theory, TQFT axiom

Lecture 15 Moduli spaces

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

Syllabus