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Physics 229: Advanced Mathematical Methods in Physics

Course Information for Fall 2016 / Winter 2017

We will learn modern geometric methods in physics.

Topics to be covered:

 Fall 2016

  • Differential Manifolds, Riemannian Manifolds

                   Lecture Note 1

  • Differential Forms

                   Lecture Note 2

  • Cohomologies

                   Lecture Note 3

  • Complex Manifolds, Kaehler Manifolds

                   Lecture Note 4

  • Vector Bundles, Gauge Theory

                   Lecture Note 5

  • Homology

                   Lecture Note 6

  • Characteristic Classes

                   Lecture Note 7

  • Supersymmetry and Index Theorems

                   Lectute Note 8

                   Note on Witten Index

                   Supersymmetry and Morse Theory

                   Supersymmetric Sigma Model

Winter 2017

  • Homotopy

                   Lecture Note 9

                   Ref: Chapter 4 of Nakahara

  • Geometry of Lie Groups

                   Lecture 10

                   Ref: Sections 5.6 and 5.7 of Nakahara

  • Geometry of Gauge Theory

                   Lecture 11

  • Random Matrix Models

                   Lecture 12         

                   Ref: Lectures on Matrix Model by M. Marino, Chapter 2
                                 Quantum Field Theory Techniques in Graphical Enumeration
                          by D. Bessis, C. Itzykson, and J.-B. Zuber
  • Chern-Simon Gauge Theory

                   Lecture 13

                   Quantum Field Theory and the Jones Polynomials by E. Witten

  1. The default grade is pass/fail since this is an advanced graduate course.
  2. I can provide letter grades on request.
  3. There will be a take home exam distributed at the last class of each quater.
  4. There will be no time limit, and you are allowed to use my lecture notes and your hand-written notes.
  5. You can expect that exam problems to be at the level of questions in my lecture notes.


Text Book:

     I will distribute my notes every week. The notes are accessible from the Caltech network.

Recommended books:

  • "Geometry, Topology and Physics" 2nd edition by M. Nakahara.
  • "Gravitation, Gauge Theories and Differential Geometry" by T. Eguchi, P. B. Gilkey and A. J. Hanson.
  • "Geometry of Differential Forms" by S. Morita.
  • "Geometrical Methods of Mathematical Physics" by B. Schutz.