Differential Geometry and Topology in Physics, Spring 2026

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

Lecture 5 Hodge theorem, Poincaré duality, Riemannian geometry

Lecture 6 Einstein equations, Maxwell equations, Symplectic geometry

Lecture 7 Arnold–Liouville theorem, Simplicial homology groups

Lecture 8 Homotopy, Cohomology groups

Lecture 9 Lefschetz fixed-point theorem, Poincaré–Hopf theorem, Abelian Chern–Simons theory

Lecture 10 Fundamental groups, Homotopy groups

Lecture 11 Lie groups, Lie algebras

Lecture 12 Vector bundles, Principal G-bundles

Lecture 13 Connections, curvatures, Yang–Mills action, Characteristic classes

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

Syllabus