UIUC-WUSTL   Joint Seminar in Symplectic and Poisson Geometry

Title: Poisson structures, Lie bialgebroids and T-duality

Abstract: In this talk we will show how the definition of Poisson algebras associated to twisted Dirac structures on smooth manifolds, on the one hand, and the description of T-duality as a duality of Lie algebroids, on the other hand, use the same type of geometric constraint imposed by a 3-form background. Particular examples of each one of these instances will be presented.  

Title: Frobenius objects in the category of spans and the symplectic category

Abstract: It is well known that Frobenius algebras are in correspondence with 2-dimensional TQFT. In this first part (as a prelude to Raj's talk in the afternoon), we introduce Frobenius objects in any monoidal category and in particular, in the category where objects are sets and morphisms are spans of sets. We prove the existence of a simplicial set that encodes the data of the Frobenius structure in this category. This serves as a (simplicial) toy model of the Wehrheim-Woodward construction for the symplectic category.

This is part of a program that intends to describe, in terms of higher category theory, the relationship between symplectic groupoids and topological field theory, via the Poisson sigma model. Based on joint work with Raj Mehta and Molly Keller (Rev. in Math. Phys (34) 10 (2022)), Raj Mehta, Adele Long and Sophia Marx (https://arxiv.org/abs/2208.14716), and ongoing work with Raj Mehta and Walker Stern.

Title: Generating functions and immersed Lagrangian Floer theory

Abstract: Following Hormander, generating functions can be used to produce an important family of immersed Lagrangians in the cotangent bundle of a manifold M and Legndrians in the 1-jet space of M. In this talk, I’ll give a report on a joint work with Kenji Fukaya where we study Lagrangain Floer homology (resp. Legendrian contact homology) of such Lagrangians (resp. Legendrian). In particular, we show that such Lagrangians admit bounding chains and we compute the Lagrangian Floer homology groups defined with respect to such bounding chains. On the Legndrian side, part of this result can be interpreted as an existence result for augmentations of Legnedrians, generalizing earlier works on 1-jet space of 1- and 2-dimensional manifolds to arbitrary dimensions. If time permits, I’ll also discuss the relationship between our results and Eliashberg-Gromov’s finite dimensional approach to Lagrangian intersections.

Title: Lifting S^1-actions to integrable systems

Abstract: An S^1-action is called Hamiltonian if it is obtained as the flow of a Hamiltonian vector field of a real-valued function J. On the other hand, an integrable system on a symplectic 4-manifold is given by a pair of independent Poisson commuting functions on the manifold. Suppose that J generates an S^1-action on a symplectic 4-manifold M. Karshon gave necessary and sufficient conditions for when this S^1-action can be extended to a Hamiltonian T^2-action. We study the following question generalizing Karshon's result: when can we find an additional function H so that (J,H) is an integrable system on M? Furthermore, how "nice" can we hope the resulting system to be? We study the properties of the resulting integrable system, and, in particular, we show that any such S^1-action can be extended to an integrable system such that all singularities are non-degenerate, except possibly for finitely many degenerate orbits of so-called parabolic type. This is joint with Sonja Hohloch. We close with some comments on generalizing these types of questions to higher dimensions.

Title: Frobenius pseudomonoids in the bicategory of spans

Abstract: I will describe a higher categorical analogue of the correspondence between simplicial sets (satisfying certain conditions) and Frobenius objects in the category of spans, described in Ivan's talk. The main result is that Frobenius pseudomonoids in the bicategory of spans correspond to paracyclic 2-Segal sets. In the talk, I will explain what all these words mean and sketch the correspondence. This is based on work in progress with Ivan Contreras and Walker Stern.