Conference 2 Schedule
Wednesday, July 19
9:00 - 10:00 Joshua Wang (Harvard University)
Title: Progress towards the colored sl(N) homology of T(2,m)
Abstract: A precise prediction for the colored sl(N) homology of any 2-bridge knot or link can be made based on a conjectural relationship with SU(N) instanton homology. It is predicted to be isomorphic to the ordinary cohomology of an associated closed manifold consisting of SU(N) representations of the fundamental group of its complement. I will explain some recent progress towards verifying the prediction for the torus knot or link T(2,m), and other applications partially based on joint work with Mike Willis.
10:15 - 11:15 Ali Daemi (Washington University)
Title: Generating functions and immersed Lagrangian Floer theory
Abstract: Following Arnold and Hörmander, generating functions can be used to produce an important family of immersed Lagrangians in the cotangent bundle of a manifold M and Legendrians in the 1-jet space of M. In this talk, I will present a report on joint work with Kenji Fukaya, where we study Lagrangian Floer homology (resp. Legendrian contact homology) of such Lagrangians (resp. Legendrians). Specifically, we demonstrate that such Lagrangians admit bounding chains and compute the Lagrangian Floer homology groups defined with respect to these bounding chains. On the Legendrian side, a portion of this result can be interpreted as the existence of a canonical augmentation for any Legendrian represented by a generating function, thus generalizing earlier works on the 1-jet space of 1- and 2-dimensional manifolds to arbitrary dimensions. As an application, we obtain topological lower bounds on the number of Reeb chords of Legendrians induced by generating functions, in the same spirit as the Arnold conjecture for Legendrians.
11:30 - 12:30 Robert Lipshitz (University of Oregon)
Title: Torsion in Legendrian contact homology
Abstract: We will start by introducing Legendrian knots and their classical invariants. We will then spend most of the talk on their first non-classical invariant, the Chekanov-Eliashberg DGA -- its definition and some of its connections to other parts of the field. We will end with some recent computations and some questions they answer and raise. The last part is joint work with Lenhard Ng, and overlaps with recent work of Frédéric Bourgeois and Salammbo Connolly.
2:30 - 3:30 Laura Starkston (UC Davis)
Title: Weinstein manifolds, multisections with divides, and Lefschetz fibrations
Abstract: We will talk about different diagrammatic ways of representing Weinstein manifolds including a new method called "multisections with divides." We will explain connections between these different representations, and potential applications. This is based on joint work with Gabriel Islambouli.
4:00 - 5:00 Rohil Prasad (UC Berkeley)
Title: Periodic points of rational area-preserving homeomorphisms
Abstract: Let S be a compact surface of genus at least 2. Any area-preserving homeomorphism F isotopic to the identity has an associated "rotation vector"; this is an element of H_1(S) which measures the average winding around S of an isotopy from the identity to F. In this talk, I'll show that if the rotation vector is a real multiple of a rational class, then F must have infinitely many periodic orbits. The proof combines results from Floer theory and topological surface dynamics.
6:00 - 8:00 BANQUET
Thursday, July 20
9:30 - 10:30 Vinicius Ramos (Matemática Pura e Aplicada) (previously scheduled for 10:45)
Title: The Toda lattice, billiards and the Viterbo conjecture
Abstract: The Toda lattice is one of the earliest examples of non-linear completely integrable systems. Under a large deformation, the Hamiltonian flow can be seen to converge to a billiard flow in a simplex. In the 1970s, action-angle coordinates were computed for the standard system using a non-canonical transformation and some spectral theory. In this talk, I will explain how to adapt these coordinates to the situation to a large deformation and how this leads to new examples of symplectomorphisms of Lagrangian products with toric domains. In particular, we find a sequence of Lagrangian products whose symplectic systolic ratio is one and we prove that they are symplectic balls. This is joint work with Y. Ostrover and D. Sepe.
10:45 - 11:45 Dan Cristofaro-Gardiner (University of Maryland) (previously scheduled for 9:30)
Title: The failure of packing stability
Abstract: A finite volume symplectic manifold is said to have ``packing stability" if the only obstruction to symplectically embedding sufficiently small balls is the volume obstruction. Packing stability has been shown in a variety of cases and it has been conjectured that it always holds. I will discuss joint work with Richard Hind giving counterexamples to this conjecture; in fact, we give examples of open and bounded domains in R^4, diffeomorphic to discs, that can not be fully packed by any domain with smooth boundary nor by any convex domain. Closely tied to our work is another old question, which asks to what extent an open symplectic-manifold has a well-defined boundary: we show that many examples for which packing stability fails can not be symplectomorphic to the interior of any compact symplectic manifold with smooth boundary. Our results can be quantified in terms of the Minkowski dimension and I will also explain this.
1:30 - 2:30 Guangbo Xu (Rutgers University)
Title: Hofer-Zehnder conjecture for toric manifolds
Abstract: (joint with Shaoyun Bai) While the Arnold conjecture provides a (topological) lower bound on the number of fixed points of Hamiltonian diffeomorphisms, it has been proved that in many cases Hamiltonian diffeomorphisms have infinitely many periodic points (i.e. fixed points of iterations), such as the proof of the Conley conjecture in various cases. On the other hand, manifolds such as projective spaces admit Hamiltonian diffeomorphisms which have only finitely many simple periodic orbits. The Hofer-Zehnder conjecture stated that even for these manifolds, if the number of fixed points is strictly greater than the Arnold lower bound, i.e., there are "redundant" fixed points, then there should be infinitely many periodic points. A recent major breakthrough towards the Hofer-Zehnder conjecture was the work of Shelukhin, who proved that for a monotone symplectic manifold, when the quantum cohomology is semisimple (over a certain field), then the Hofer-Zehnder conjecture is true. Inspired by Shelukhin's work and the picture of mirror symmetry, we prove the Hofer-Zehnder conjecture for ALL compact toric manifolds. There are a few key ingredients in the proof. First, following Givental, Hori-Vafa, Fukaya-Oh-Ohta-Ono etc. toric manifolds are mirror to Landau-Ginzburg models and hence their quantum cohomology are "generically" semisimple. Second, as toric manifolds are GIT quotients of vector spaces, we use the gauged linear sigma model (GLSM) to do Floer theory over integers without the need of virtual technique.
3:00 - 4:00 Daniel Alvarez-Gavela (MIT)
Title: The nearby Lagrangian conjecture: past, present and future
Abstract: The nearby Lagrangian conjecture states that any exact closed connected Lagrangian submanifold of the cotangent bundle of a closed connected smooth manifold is Hamiltonian isotopic to the zero section. In particular the conjecture predicts that any such Lagrangian is diffeomorphic to the zero section. I will review what is known about the problem, including some recent progress joint with M. Abouzaid, S. Courte and T. Kragh. In an attempt to not be completely dissonant with the conference name I will also make some remarks about connections with pseudo-holomorphic curves as well as discuss some special phenomena which occur in low dimensions.
Friday, July 21
9:30 - 10:30 Jo Nelson (Rice University)
Title: Torus knotted Reeb dynamics in the tight 3-sphere
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.
10:45 - 11:45 Andy Manion (North Carolina State University)
Title: A cornered gluing operation for DA bimodules
Abstract: After some context and motivation, I will discuss a new algebraic gluing operation that should relate to CFDA(H) for bordered sutured Heegaard diagrams H in the same way that higher tensor products relate to A(Z) for arc diagrams Z. I will sketch a proposed "cornered Heegaard Floer gluing formula" for CFDA bimodules based on this new algebraic operation.
1:30 - 2:30 Luya Wang (UC Berkeley)
Title: A connected sum formula for embedded contact homology
Abstract: The contact connected sum is a well-understood operation for contact manifolds. I will focus on the 3-dimensional case and the Weinstein 1-handle model for the contact connected sum. I will discuss how pseudo-holomorphic curves in the symplectization behave under this operation. After reviewing embedded contact homology, we will see how this results in a chain-level description of the embedded contact homology of a connected sum.
3:00 - 4:00 John Etnyre (Georgia Tech)
Title: Symplectic embeddings of rational homology balls into projective space
Abstract: I will discuss how to builds small symplectic caps for contact manifolds and in particular give a handlebody descriptions of symplectic embeddings of rational homology balls into CP^2. This is joint work with Hyunki Min, Lisa Piccirillo, and Agniva Roy.