Tag - Geometry

Michael Hutchings: How not to define cylindrical contact homology

18th October, 2013

We consider the problem of defining cylindrical contact homology, in the absence of contractible Reeb orbits, using "classical" methods. The main technical difficulty is failure of transversality of multiply covered cylinders. One can fix this difficulty by using S1-dependent almost complex structures, but at the expense of introducing another difficulty which we will explain. We outline how fixing the latter difficulty ultimately leads to a different theory, an analogue of positive symplectic homology. This talk is intended to be part of a series of expository talks on the foundations of contact homology, but prerequisites should be minimal.

Peter Weigel: Positive loops and orderability in contact geometry

4th October, 2013

Orderability of contact manifolds is related in some non-obvious ways to the topology of a contact manifold Σ. We know, for instance, that if Σ admits a 2-subcritical Stein filling, it must be non-orderable. By way of contrast, in this talk I will discuss ways of modifying Liouville structures for high-dimensional Σ so that the result is always orderable. The main technical tool is a Morse-Bott Floer-theoretic growth rate, which has some parallels with Givental's non-linear Maslov index. I will also discuss a generalization to the relative case, and applications to bi-invariant metrics on Cont(Σ).

Dietmar Salamon: GIT and μ-GIT

26th September, 2013

In this lecture I will explain the moment-weight inequality, and its role in the proof of the Hilbert-Mumford numerical criterion for μ-stability. The setting is Hamiltonian group actions on closed Kaehler manifolds. The major ingredients are the moment map μ and the finite-dimensional analogues of the Mabuchi functional and the Futaki invariant.

Richard Kenyon: Dimers and Integrability

29th March, 2013

To any convex integer polygon we associate a Poisson variety, which is essentially the moduli space of connections on line bundles on (certain) bipartite graphs on a torus. There is an underlying integrable Hamiltonian system whose Hamiltonians are weighted sums of dimer covers.

Nancy Hingston: Resonance for Loop Homology of Spheres

15th March, 2013

Fix a metric (Riemannian or Finsler) on a compact manifold M. The critical points of the length function on the free loop space 𝓛M of M are the closed geodesics on M. Filtration by the length function gives a link between the geometry of closed geodesics, and the algebraic structure given by the Chas-Sullivan product on the homology of 𝓛M and the 'dual' loop cohomology product.

If X is a homology class on 𝓛M, the 'minimax' critical level Cr(X) is a critical value of the length function. Gromov proved that if M is simply connected, there are positive constants k and K so that for every homology class X of degree > dim(M) on 𝓛M, k < deg(X)/Cr(X) < K. When M is a sphere, we prove there are positive constants a and b so that for every homology class X on 𝓛M, aCr(X)-b < deg(X) < aCr(X)+b. There are interesting consequences for the length spectrum.

Alvaro Pelayo: Intermediate Symplectic Capacities

1st March, 2013

In 1985 Misha Gromov proved his Nonsqueezing Theorem, and hence constructed the first symplectic 1-capacity. In 1989 Helmut Hofer asked whether symplectic d-capacities exist if 1 < d < n. I will discuss the answer to this question and its relevance in symplectic geometry.