### Aim of the Course

To introduce some modern results in geometric analysis. Particular goals are to discuss the following topics:

- The positive mass/energy theorem in Lorentzian and Riemannian geometry;
- The use of the positive mass theorem to solve the Yamabe problem in low dimensions;
- The relationship between quantities that arise in the Yamabe problem and functionals introduced by Perelman in his proof of the Poincare conjecture.

It may turn out that the above plan is too ambitious, in which case some of the material may need to be dropped.

### Office hours

I will be in my office, C 237, Monday 9-10am. You may drop by then to ask questions. If you wish to see me at any other time, please email me so that we can make an appointment.

### Sources

The course should be quite self-contained, and I will make a set of typed notes available as the course progresses. Most of the material is quite modern, and cannot be found in a single textbook. The initial parts on the positive mass/energy theorem will follow the paper of Parker and Taubes:

T. Parker and C.H. Taubes, **On Witten's proof of the positive energy theorem**, Comm. Math. Phys. **84**, 223-238 (1982).
This article, in turn, is based Witten's original article:

E. Witten, **A new proof of the positive energy theorem**, Comm. Math. Phys. **80**, 381-402 (1981).
More details concerning Clifford algebras and some details of Witten's proof of the positive mass/energy theorem may be found in

H.B. Lawson and M.-L. Michelsohn, **Spin Geometry** (Princeton University Press, Princeton, 1989).
Details on the Yamabe problem may be found in

T. Aubin, **Some Nonlinear Problems in Riemannian Geometry** (Springer Verlag, Berlin, 1998),
although this is not really a place that I would recommending learning this (or any other) material from. There is also the review article

J.M. Lee and T.H. Parker, **The Yamabe problem**, Bull. A.M.S. **17**, 37-91 (1987),
although this assumes a fair amount of background knowledge.

The connection between the Yamabe problem and Perelman functional was pointed out in the paper

K. Akutagawa, M. Ishida, C. LeBrun, **Perelman's Invariant, Ricci Flow, and the Yamabe Invariants of Smooth Manifolds**, Arch. Math. **88**, 71-76 (2007).
The original paper by Perelman may be found at

G. Perelman, **The entropy formula for the Ricci flow and its geometric applications**.
There are now various papers, books and websites dedicated to expanding and explaining Perelman's papers on the Geometrisation Conjecture. A good place to start, if you want to find out more about this topic, is the Kleiner-Lott website.