Joint IAS-PU Symplectic Geometry Seminar

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.

Symplectic fillings and star surgery

Laura Starkston
University of Texas, Austin
September 25, 2014
Although the existence of a symplectic filling is well-understood for many contact 3-manifolds, complete classifications of all symplectic fillings of a particular contact manifold are more rare. Relying on a recognition theorem of McDuff for closed symplectic manifolds, we can understand this classification for certain Seifert fibered spaces with their canonical contact structures. In fact, even without complete classification statements, the techniques used can suggest constructions of symplectic fillings with interesting topology.

Measures on spaces of Riemannian metrics

Dmitry Jakobson
McGill University
July 21, 2014
This is joint work with Y. Canzani, B. Clarke, N. Kamran, L. Silberman and J. Taylor. We construct Gaussian measure on the manifold of Riemannian metrics with the fixed volume form. We show that diameter and Laplace eigenvalue and volume entropy functionals are all integrable with respect to our measures. We also compute the characteristic function for the \(L^2\) (Ebin) distance from a random metric to the reference metric.

Minimal Discrepancy of Isolated Singularities and Reeb Orbits

Mark McLean
Stony Brook University
April 4, 2014
Let A be an affine variety inside a complex N dimensional vector space which either has an isolated singularity at the origin or is smooth at the origin. The intersection of A with a very small sphere turns out to be a contact manifold called the link of A. Any contact manifold contactomorphic to the link of A is said to be Milnor fillable by A. If the first Chern class of our link is 0 then we can assign an invariant of our singularity called the minimal discrepancy, which is an important invariant in birational geometry.

Princeton/IAS Symplectic Geometry Seminar

Keon Choi
University of California, Berkeley
March 7, 2014
Embedded contact homology is an invariant of a contact three-manifold, which is recently shown to be isomorphic to Heegaard Floer homology and Seiberg-Witten Floer homology. However, ECH chain complex depends on the contact form on the manifold and the almost complex structure on its symplectization. This fact can be used to extract symplectic geometric information (e.g. ECH capacities) but explicit computation of the chain complexes has been carried out only on a few cases.