Joint IAS-PU Symplectic Geometry Seminar

Disc filling and connected sum

Kai Zehmisch
Universität Münster
February 27, 2015
In my talk I will report on recent work with Hansjörg Geiges about a strong connection between the topology of a contact manifold and the existence of contractible periodic Reeb orbits. Namely, if the contact manifold appears as non-trivial contact connected sum and has non-trivial fundamental group or torsion-free homology, then the existence is ensured. This generalizes a result of Helmut Hofer in dimension three.

The symplectic displacement energy

Peter Spaeth
GE Global Research
February 20, 2015
To begin we will recall Banyaga's Hofer-like metric on the group of symplectic diffeomorphisms, and explain its conjugation invariance up to a factor. From there we will prove the positivity of the symplectic displacement energy of open subsets in compact symplectic manifolds, and then present examples of subsets with finite symplectic displacement energy but infinite Hofer displacement energy. The talk is based on a joint project with Augustin Banyaga and David Hurtubise.

Symplectic homology via Gromov-Witten theory

Luis Diogo
Columbia University
February 13, 2015
Symplectic homology is a very useful tool in symplectic topology, but it can be hard to compute explicitly. We will describe a procedure for computing symplectic homology using counts of pseudo-holomorphic spheres. These counts can sometimes be performed using Gromov-Witten theory. This method is applicable to a class of manifolds that are obtained by removing, from a closed symplectic manifold, a symplectic hypersurface of codimension 2. This is joint work with Samuel Lisi.

Symplectic forms in algebraic geometry

Giulia Saccà
Member, School of Mathematics
January 30, 2015
Imposing the existence of a holomorphic symplectic form on a projective algebraic variety is a very strong condition. After describing various instances of this phenomenon (among which is the fact that so few examples are known!), I will focus on the specific topic of maps from projective varieties admitting a holomorphic symplectic form. Essentially, the only such maps are Lagrangian fibrations and birational contractions. I will motivate why one should care about these types of maps, and give many examples illustrating their rich geometry.

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.

\(C^0\)-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 \((M, \omega)\).

Designing low energy capture transfers for spacecraft to the Moon and Mars

Edward Belbruno
Princeton University and Innovative Orbital Design, Inc.
October 28, 2014
In 1991 a new type of transfer to the Moon was operationally demonstrated by the Japanese spacecraft, Hiten, using ballistic capture. It was designed by this speaker and James Miller. This is capture about the Moon which is automatic so that no rocket engines are required. It was accomplished due to the existence of regions in phase space called weak stability boundaries, where ballistic capture occurs. These are complex fractal regions of unstable chaotic motion. Until recently it was thought that such a transfer to Mars was not feasible.

Equivariant structures in mirror symmetry

James Pascaleff
University of Illinois at Urbana-Champaign
October 17, 2014
When a variety \(X\) is equipped with the action of an algebraic group \(G\), it is natural to study the \(G\)-equivariant vector bundles or coherent sheaves on \(X\). When \(X\) furthermore has a mirror partner \(Y\), one can ask for the corresponding notion of equivariance in the symplectic geometry of \(Y\). The infinitesimal notion (equivariance for a single vector field) was introduced by Seidel and Solomon (GAFA 22 no. 2), and it involves identifying a vector field with a particular element in symplectic cohomology.