Math 865 Advanced Topics in Geometry (Fall 2011)
Lectures are by
Jeff Viaclovsky on Tuesdays and Thursdays
at 02:30-03:45 PM in Van Vleck B131.
- Riemannian Geometry by Peter Petersen.
- Differential Geometric Structures by Poor.
- Einstein Manifolds by Besse.
- Lecture notes in Geometric Analysis (Old
notes, see new ones below).
The class mailing list is math865-1-f11 "at" lists.wisc.edu.
NOTE: We did not get to the last 2 topics since we did more
complex geometry than initally planned.
These topics will be discussed next semester in
the PDE Course 820.
- First few lectures will be a quick review of tensor calculus
and Riemannian geometry: metrics, connections, curvature tensor, Bianchi identities,
commuting covariant derivatives, etc.
- Decomposition of curvature tensor into irreducible summands.
- Bochner-Weitzenbock formulas: various curvature conditions yield topological restrictions on a manifold.
- Review of elliptic theory in Holder and Sobolev spaces.
Theory of elliptic operators on Riemannian manifolds with basic Fredholm Theory,
with applications to Hodge Theory.
- On non-compact manifolds we will consider Fredholm operators on weighted spaces,
such as weighted Sobolev and Holder spaces. This has applications to the study of
asymptotically locally Euclidean spaces (ALE) spaces, such as the Eguchi-Hanson metric.