Differential Geometry and Topology in Physics, Spring 2019

Lecture 1 Euler characteristics and supersymmetric quantum mechanics

Lecture 2 Manifolds, tangent spaces

Lecture 3 Vector fields, tangent bundles, orientation

Lecture 4 Cotangent bundles, differential forms

Lecture 5 de Rham cohomology, metric, harmonic forms, Hodge theorem, Poincaré duality

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

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

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