Geometric Analysis

Goals: This course is an introduction to geometric analysis on Riemannian manifolds. It introduces some of the important geometric PDEs on manifolds, including examples of linear and non-linear elliptic equations and linear and non-linear parabolic (i.e. heat flow) equations. The emphasis is on quickly acquiring a working knowledge of the tools and ideas of the subject.

Prerequisites: Familiarity with manifolds (vector fields, differential forms, tangent and tensor bundles). It will be very useful, but not strictly necessary, to have an understanding of Hilbert and Banach spaces (e.g. Chapters 3-5 of Rudin's ``Real and Complex Analysis'') and some knowledge of PDEs.

Homework:   HW1    HW2    HW3    HW4

Lecture Notes:   Primer: Sections 1-4    Direct Methods  

Recommended Text: Lectures on the Geometry of Manifolds by Liviu Nicolaescu.      Course outline

Preliminary list of topics:

1. Connections and curvature on vector bundles.
2. A working man's introduction to elliptic theory.
3. Finding geodesics via Morse theory.
4. The Hodge Theorem and the Bochner technique.
5. Spinors and the Dirac equation.
6. The Seiberg-Witten equations.
7. The heat kernel on manifolds.
8. Time permitting: pseudo-holomorphic maps or harmonic map heat flow.

Other helpful reference books:

• The Laplacian on Riemannian Manifolds by Steven Rosenberg.
• Non-linear Analsis on Manifolds. Monge-Ampere Equations by Thierry Aubin.
• Elliptic Partial Differential Equations by D.Gilbarg and N.Trudinger.
• J-Holomorphic Curves and Symplectic Geometry by D.McDuff and D. Salomon