Math 259C (Geometrical Physics)

Course Topics: Mathematical and Numerical General Relativity
Instructor: Prof. Michael Holst (5739 AP&M,
Term: Winter 2009
Lecture: 9:30a-10:50a Tu-Th, APM 7421
Class webpage:
  • M. Alcubierre, Introduction to 3+1 Numerical Relativity, Oxford, 2008.
  • T. Frankel, The Geometry of Physics, Cambridge, 2004, second edition.

CATALOG DESCRIPTION: 259A-B-C. Geometrical Physics (4-4-4)
Manifolds, differential forms, homology, deRham’s theorem. Riemannian geometry, harmonic forms. Lie groups and algebras, connections in bundles, homotopy sequence of a bundle, Chern classes. Applications selected from Hamiltonian and continuum mechanics, electromagnetism, thermodynamics, special and general relativity, Yang-Mills fields. Prerequisite: graduate standing in mathematics, physics, or engineering, or consent of instructor.

GRADES, EXAMS, DATES: Your grade in the course is based on attending the lectures and participating in the class.

ANNOUNCEMENTS, NOTES, ETC: Books, papers, and notes that I will steal from throughout the quarter:

SCHEDULE FOR THE LECTURE TOPICS: The overall plan is to build some background in geometric PDE and mathematical general relativity, and then develop mathematical techniques in nonlinear approximation to build provably good numerical methods for solving the various types of elliptic and evolution PDE systems that arise. Our focus will be to understand what the key mathematical and approximation issues and obstacles are in the study of the constraint and evolution equations in general relativity. We will also examine some related problems in geometric flows along the way.

We will make the course as self-contained as possible, which will require that we develop some basic knowledge in four areas:

  • Background: Geometric Mechanics, Geometric PDE, Sobolev Spaces on Manifolds
  • Mathematical General Relativity (Evolution Equations and Constraints)
  • Numerical Methods and Approximation Theory for Geometric Elliptic PDE
  • Numerical Methods and Approximation Theory for Geometric Evolution PDE
Above I have listed the main sources (books, paper, notes) that I will use to develop some of the material in the lectures. The specific topics for the lectures will evolve as the quarter proceeds, based on how the interests of the participants and the lecturer evolve.