CU Graduate Mathematics Society

Refer to the overview page for room numbers and details of other talks in the seminar series.

9:05 Christopher Braun (cdb35) Anti-gravity Chocolate Syrup

Morse homology is an important prototype for its big brother, Floer homology. Crudely: given a suitably nice function on a manifold, we look at the gradient field. The critical points generate our chains, and our boundary map counts the flow lines between them. Morse homology ends up being canonically isomorphic to singular homology.
I will define the machinery of Morse homology (with Floer homology in mind), and provide lots of examples, then say something (but not much) about the isomorphism with singular homology. This talk should complement the other talks on Floer homology and provide the necessary introduction. As a result it should be easy to follow.
There will be several memorable practical demonstrations during the talk (and also a chance for those attending to try them).

10:00 Rune Haugseng Constructing Floer homology (on the cotangent bundle)

On a symplectic manifold, associated to a smooth periodic function (or "Hamiltonian") H we have the Hamiltonian equation dx/dt = XH(x) (where XH is the "symplectic gradient" of H). In good cases, we can construct a chain complex generated by periodic solutions to this equation, and Floer homology is the homology of this complex. I will try to explain how this is done in the case of cotangent bundles, in particular how we get a boundary map by studying the space of solutions to another differential equation that tend asymptotically to solutions of the Hamiltonian equation. Time permitting, I will also say a few words about how you prove that this space has the properties required for the construction to work.

12:05 Luis Haug (lsh31) Floer homology for cotangent bundles and loop space homology

Floer Homology was first defined for compact symplectic manifolds and later extended to cotangent bundles T*M. In this case it is isomorphic to the singular homology of the loop space of M. This result has been proved in three different ways, most recently by A. Abbondandolo and M. Schwarz.

Their approach makes use of the fact that the Hamiltonian function H on T*M whose action functional is used to construct the Floer complex can be chosen arbitrarily within a suitable class. In particular, the Legendre transform of a nice Lagrangian function L on TM will do. The Lagrangian action functional associated to L is better behaved than the one for H, and it is possible to construct a Morse complex for it, whose homology is the singular homology of the loop space.

The isomorphism proposed by Abbondandolo and Schwarz constructs a chain complex isomorphism between the Morse complex and the Floer complex using methods adapted from Floer theory.

I will talk about the ideas and techniques used in the construction of this isomorphism. I'll do my best to keep the talk as self-contained as possible, but it would probably be a good idea for people interested in it to go also to one of the other talks on Floer homology.

14:00 Will Merry (wjm29) The Boundary rigidity problem

I will talk about a problem that takes its motivation from geophysics, namely, can we determine what the centre of the earth is like by studying  the surface? More precisely, given a Riemannian manifold (M,g) with boundary, if we know the geodesic distance between any two points on the boundary, can we reconstruct what the metric looks like in the interior?

Phrased in this generality, the answer is resoundingly `no', since if F:M-->M is a diffeomorphism of M which is the identity when restricted to the  boundary then the pullback metric F*g is easily seen to have the same distance function as our original metric restricted to the boundary.

This in some sense is the natural obstruction to the problem, and the question becomes much more interesting when we ask 'can we determine the metric up to such a diffeomorphism F'? Michel conjectured in 1981 that for a simple closed manifold the answer is yes, and I will outline the progress made towards this over the last 25 years.

Finally I will discuss some very recent work concerning a generalisation of the problem, which involves adding a magnetic field into the mix.

The talk will be elementary - if you attended differential geometry last term then you'll be fine, and, at a pinch any brave part II student who's taken part II differential geometry will be able to follow it. In particular, I won't assume you know what a 'geodesic' is.