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)