Princeton University

Symplectic embeddings from concave toric domains into convex ones

Dan Cristofaro-Gardiner
Harvard University
October 24, 2014
Embedded contact homology gives a sequence of obstructions to four-dimensional symplectic embeddings, called ECH capacities. These obstructions are known to be sharp in several interesting cases, for example for symplectic embeddings of one ellipsoid into another. We explain why ECH capacities give a sharp obstruction to embedding any "concave toric domain" into a "convex" one. We also explain why the ECH capacities of any concave or convex toric domain are determined by the ECH capacities of a corresponding collection of balls.

An algebro-geometric theory of vector-valued modular forms of half-integral weight

Luca Candelori
Lousiana State University
October 23, 2014
We give a geometric theory of vector-valued modular forms attached to Weil representations of rank 1 lattices. More specifically, we construct vector bundles over the moduli stack of elliptic curves, whose sections over the complex numbers correspond to vector-valued modular forms attached to rank 1 lattices. The key idea is to construct vector bundles of Schrodinger representations and line bundles of half-forms over appropriate `metaplectic stacks' and then show that their tensor products descend to the moduli stack of elliptic curves.

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.

The standard \(L\)-function for \(G_2\): a "new way"

Nadya Gurevich
Ben-Gurion University of the Negev
October 2, 2014
We consider a Rankin-Selberg integral representation of a cuspidal (not necessarily generic) representation of the exceptional group \(G_2\). Although the integral unfolds with a non-unique model, it turns out to be Eulerian and represents the standard \(L\)-function of degree 7. We discuss a general approach to the integrals with non-unique models. The integral can be used to describe the representations of \(G_2\) for which the (twisted) \(L\)-function has a pole as functorial lifts. This is a joint work with Avner Segal.

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.

Gamma Ray Bursts from a Different Angle: The Sequel

David Eichler
Ben Gurion University
September 23, 2014

A classic problem posed by long gamma ray bursts (GRB) is that the energy output requires gravitational energy release so deep within the host star that the prompt gamma rays should, upon naive consideration, have been obscured. It is suggested that photons emitted along the direction of the emitting plasma's motion are indeed geometrically blocked by optically thick baryonic matter, and that we usually see the photons that are emitted nearly backward in the frame of the emitting plasma. Many puzzling observations concerning GRB then fall into place.