Differential geometry and topology in Physics, Spring 2017

Syllabus

Lecture notes

lecture1 (Euler characteristics and supersymmetric quantum mechanics [hand-written])

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

lecture3 (cotangent bundles, differential forms)

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

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

lecture6 (homotopy, fundamental groups, homotopy groups)

lecture7 (simplicial complex, homology, Lefschetz fixed-point theorem, Poincare-Hopf theorem)

lecture8 (Lie groups, Lie algebras, vector bundles)

lecture9 (principal G-bundles, connections, curvatures, Yang-Mills action)

lecture10 (characteristic classes, Chern-Weil theory)

lecture11 (flat connections, Chern-Simons theory)

lecture12 (index theorem, Hirzebruch-Riemann-Roch theorem, anomaly, supersymmetry)

Epilogue (quantum Hall effects, this note is essentially a write-up of Witten's slides)

Extra1 (complex, Kahler, Calabi-Yau manifolds)

Extra2 (symplectic geometry, integrable system)

(These notes are written by referring to various sources without mentioning them. Comments are welcome.)

Added: There are many incorrect statements, and updated notes are available for the course in 2019.