Differential Geometry and Topology in Physics, Spring 2017

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

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

Syllabus