Millican Colloquium: Conformal Geometry, Holography, and Corners -- Stephen McKeown (Princeton) | Department of Mathematics

Millican Colloquium: Conformal Geometry, Holography, and Corners -- Stephen McKeown (Princeton)

Event Information
Event Location: 
GAB 461 (Refreshments at 3:30 in 472)
Event Date: 
Monday, November 21, 2016 - 4:00pm

Conformal geometry is the branch of differential geometry that studies what we can say about a geometric space if all we know is how to measure angles between crossing curves, but not lengths. Specifically, it is interested in what does and does not change if you scale lengths but not angles, and what problems can be solved in the process. An elementary example of the relevance of conformal geometry comes from considering maps of the Earth: a map of any part of the Earth, drawn on flat paper, must distort lengths; however, it need not distort angles. We say, therefore, that there is a conformal map (or function) between the plane and neighborhoods on the sphere.

Traditionally, and in contrast to the case in Riemannian geometry, it has been very hard to find conformal invariants of a space. Most of the obvious geometric quantities one can compute, such as curvature, do not transform nicely if one rescales lengths in a smoothly varying way. There were a handful of classically known objects - the Weyl tensor, the Bach tensor in dimension four, the Cotton tensor in dimension 3, and a few others - but there was no general theory. In the 1980s, Fefferman and Graham introduced a new method for studying conformal invariants. Motivated by the example of hyperbolic space (specifically the Poincaré ball model), they initiated a program of relating the purely Riemannian geometry of a complete Einstein metric on one space and the conformal geometry of its boundary at infinity. The study of this relationship has come to be known as holography, since it encodes geometric information in $n + 1$ dimensions in the (conformal) geometry of a space of only $n$ dimensions. The project has been extremely fruitful for studying conformal geometry, and has also attracted great interest in theoretical physics, where it forms the geometric basis for the so-called AdS/CFT correspondence.

We will briefly describe Riemannian geometry, conformal geometry, and the relationship between the two. We will then describe the Fefferman-Graham program and some of its methods and successes. Finally, we will present new work that generalizes this program to the case of conformal geometry on manifolds with boundaries, where in this setting, the Einstein space has a corner at infinity. While some familiarity with manifolds will increase intuition for the methods and approach used in the talk, the talk should be accessible to students with some background in point-set topology and multivariable calculus.