Geometric flows in Mathematics and Physics (08w5110)

Objectives

There is little doubt of the timeliness of this proposal. Geometric flows have experienced quite remarkable growth recently, in no small part due to Perelman's completion of Hamilton's Ricci flow programme for 3-manifolds, and this has not gone unnoticed in the theoretical physics community, where applications abound. A first meeting to explore some of these connections will be held at the Albert Einstein Institute in Potsdam in November 2006. We seek to build on the momentum and to hold at BIRS a meeting with a broader spectrum of mathematicians and physicists working in relevant areas.

Our first objective is to expose physicists to the recent developments in geometric flow, and conversely to expose mathematicians to some of the problems physicists think could be addressed using geometric flows. A certain proportion of the talks, perhaps a third, will review the most recent mathematical progress or the most recent physical questions.

Our second objective is to stimulate the process of answering a few of these questions. To that end, another portion of talks, say again one-third, will focus on a small number of the most actively pursued questions in the time leading up to the meeting, reporting recent progress.

The remaining speaking slots will be distributed to those who raise interesting new questions or find new theorems too recent to have been exploited in applications.

We will try to limit our schedule to about 4 or 5 talks per day, some an hour long and some a half-hour long. We intend to keep about half of the working hours free for discussions and collaboration among the participants.

Among the issues the workshop will address are the following:

To what extent are Ricci flow dynamics and Einstein dynamics comparable? This is a very broad question, ranging from Gibbons’s observation that SO(n)-invariant Ricci flows are equivalent to self-dual Ricci-flat metrics in dimension n+1, to Garfinkle and Isenberg’s suggestion that singularity formation in Ricci flow on ``corsetted 3-spheres’’ has features in common with critical collapse in black hole formation in general relativity.

What role has the flow of static metrics to play? In a recent thesis, List has presented a simplified version of Ricci flow which is taylored to evolve metrics that have an isometry (such as a timelike Killing vector field). The fixed points are solutions of the static Einstein equations. Can this flow be used to construct special static metrics, such as those that minimize ADM mass within a given class? This would address a number of open problems in mathematical relativity.

What can be said of string dynamics in tachyon condensation? It is believed that closed string tachyon condensation can be described by renormalization group flow equations. The lowest order approximation to these equations is the Ricci flow, but there is a sequence of better approximations containing curvature terms of successively higher order. Can the study of these equations yield useful physics?

Following the theme of Perelman’s work on Ricci flow, is renormalization group flow a gradient flow? This amounts to a kind of C-theorem, along the lines that Zamolodchikov found in the 1980s in the context of 2-dimensional quantum field theory. Is there a C-theorem in 4 or other dimensions, and are geometric flows a tool to help find it?