|Course Title:||Riemannian Geometry|
|Course Number:||MATH 6170 - 1|
|Days:||M† 3:05-3:55 in LCB 215; W 3:05-4:55 in JWB 308. († no class on the first Mondays of the month 10-1, 11-5, 12-3.)|
|Office Hours:||in JWB 224.|
|Prerequisites:||Some knowledge of differentiable manifolds would be useful (e.g. MATH 6510-6520 or consent of instructor.)|
|Text:||Isaac Chavel, Riemannian Geometry -- A Modern Introduction, 2nd edition, Cambridge 2006|
Math 6170 - - Fall 2005 homepage.
This course is useful for students of geometry, topology, geometric group theory, nonlinear analysis, general relativity and graphic design.
Abstract. Perelman's results on Hamilton's Program to solve Thurston's Geometriztion Conjecture, which implies the Poincare Conjecture, has the mathematical world extremely excited. Mathematicians expect that detailed proofs for Perelman's arguments will eventually be given.
The objective of this course is to start from basic notions from Riemannian Geometry and build to Perelman's Theorems. How does curvature influence the local and global geometry and topology of a manifold? We shall discuss basic examples such as hyperbolic space and Riemannian surfaces in considerable detail. An important tool is to use geodesic curves, which locally minimize length and whose behavior is influenced by curvature. We shall develop the intrinsic, classical and differential form notations in parallel. We shall develop some geometric analysis tools to study the Ricci Flow, a systematic deformation that splits a manifold into understandable pieces, used by Perelman.
Topics. We shall loosely follow Chavel's text for the bulk of the course. Topics include (depending on time and student's interests):
Last updated: 9 / 14 / 7