Princeton University

Cyclic homology and \(S^1\)-equivariant symplectic cohomology

Sheel Ganatra
Stanford University
November 21, 2014
In this talk, we study two natural circle actions in Floer theory, one on symplectic cohomology and one on the Hochschild homology of the Fukaya category. We show that the geometric open-closed string map between these two complexes is \(S^1\)-equivariant, at a suitable chain level. In particular, there are induced maps between equivariant homology theories, natural with respect to Gysin sequences, which are isomorphisms whenever the non-equivariant map is.

Weyl-type hybrid subconvexity bounds for twisted L-functions and Heegner points on shrinking sets

Matthew Young
Texas A & M University; von Neumann Fellow, School of Mathematics
November 20, 2014
One of the major themes of the analytic theory of automorphic forms is the connection between equidistribution and subconvexity. An early example of this is the famous result of Duke showing the equidistribution of Heegner points on the modular surface, a problem that boils down to the subconvexity problem for the quadratic twists of Hecke-Maass L-functions. It is interesting to understand if the Heegner points also equidistribute on finer scales, a question that leads one to seek strong bounds on a large collection of central values.

\(C^0\)-characterization of symplectic and contact embeddings

Stefan Müller
University of Illinois at Urbana-Champaign
November 7, 2014
Symplectic and anti-symplectic embeddings can be characterized as those embeddings that preserve the symplectic capacity (of ellipsoids). This gives rise to a proof of \(C^0\)-rigidity of symplectic embeddings, and in particular, diffeomorphisms. (There are many proofs of rigidity of symplectic diffeomorphisms, but all known proofs of rigidity of symplectic embeddings seem to use capacities.) This talk explains a characterization of symplectic embeddings via Lagrangian embeddings (of tori); the corresponding formalism is called the shape invariant (discovered by J.-C.

On the Gromov width of polygon spaces

Alessia Mandini
University of Pavia
October 31, 2014
After Gromov's foundational work in 1985, problems of symplectic embeddings lie in the heart of symplectic geometry. The Gromov width of a symplectic manifold \((M, \omega)\) is a symplectic invariant that measures, roughly speaking, the size of the biggest ball we can symplectically embed in \((M, \omega)\).

Beyond ECH capacities

Michael Hutchings
University of California, Berkeley
October 24, 2014
ECH (embedded contact homology) capacities give obstructions to symplectically embedding one four-dimensional symplectic manifold with boundary into another. These obstructions are known to be sharp when the domain is a "concave toric domain" and the target is a "convex toric domain" (see previous talk). However ECH capacities often do not give sharp obstructions, for example in many cases when the domain is a polydisk.