MATH 568- RIEMANNIAN GEOMETRY
Course announcement for Spring 2009
Topic: Geometric concepts of mass in General Relativity and Riemannian
Geometry.
After an introduction to Lorentzian geometry (including the classical
Hawking and Penrose singularity
theorems), the plan is to introduce the background
to the following topics of recent/current research interest: (i) ADM
mass of asyptotically flat manifolds
(Bartnik) (ii) the Riemannian positive mass theorem: minimal surface proof
(Schoen-Yau), spinor proof (Witten-Taubes). (iii) The Penrose inequality
via inverse mean curvature flow (Huisken-Ilmanen)
(iv) The inequality for multiple horizons (Bray) (v) mass and the
isoperimetric profile (Huisken)
Of course, in many cases the details of proofs will not be given- the
goal is to understand
the background, statement, and techniques involved in the proofs. Prior
exposure to Riemannian geometry
helps, but I'll review what is needed in the beginning (in the
Lorentzian setting). Some PDE would help, too,
but this won't be the emphasis. The main prerequisites are experience in
graduate-level math and
curiosity about interesting (if challenging) recent developments in
geometric analysis.
Grading: based on problem sets.
Confirmation that the course will run will be based on enrollment as of
12/02
Alex Freire
Remark: did not run (one interested student)