Monday
Generating functions and immersed Lagrangian Floer theory
- Speaker: Aliakbar Daemi
- Time: 9:30-10:30
- Abstract: Following Arnold and 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. Legendrians). 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. As an application of these results, one obtains topological lower bounds on the number of Reeb chords of Legendrians induced by generating functions, in the same sprit as Arnold conjecture for Legendrians.
Torus knotted Reeb dynamics in the 3-sphere
- Speaker: Jo Nelson
- Time: 11:00 - 12:00
- Abstract: I will discuss work in progress with Morgan Weiler on the Calabi invariant of periodic orbits of symplectomorphisms of Seifert surfaces of T(p,q) torus knots in the standard contact 3-sphere. Our results come by way of spectral invariants of embedded contact homology, which allows us to realize the relationship between the action and linking of Reeb orbits with respect to an elliptic T(p,q) orbit in the standard 3-sphere. Along the way, we developed new methods for understanding the embedded contact homology of open books and prequantization orbi-bundles.
How to count curves in symplectic geometry
- Speaker: Guangbo Xu
- Time: 2:00-3:00
- Abstract: It is a common theme in symplectic geometry and gauge theory that by counting solutions to certain nonlinear elliptic equations one obtains invariants of manifolds. In symplectic geometry one obtains the famous Gromov-Witten invariants. As the moduli spaces are orbispaces, the counts are in general rational numbers. In a recent joint work with Shaoyun Bai, we developed an idea of Fukaya and Ono which allows us to obtain refined integer counts. As applications we define integer-valued Gromov-Witten invariants and proved an integral version of the Arnold conjecture. In this talk I will sketch the basic construction and potential further applications.
Homology cobordism and the geometry of hyperbolic three-manifolds
- Speaker: Francesco Lin
- Time: 3:30-4:30
- Abstract: A major challenge in the study of the structure of the three-dimensional homology cobordism group is to understand the interaction between hyperbolic geometry and homology cobordism. In this talk, I will discuss how monopole Floer homology can be used to study some basic properties of certain subgroups of the homology cobordism group generated by hyperbolic homology spheres satisfying some natural geometric constraints.
On Bennequin type inequality for symplectic caps of (S^3, \xi_std)
- Speaker: Anubhav Mukherjee
- Time: 4:40-5:40
- Abstract: In this talk I will discuss a Bennequin type inequality for symplectic caps of S^3 with standard contact structure. This has interesting applications which can help us to understand the smooth topology of symplectic caps and smoothly embedded suraces inside this. This is a joint work with Masaki Taniguchi and Nobuo Iida.
