Schedule:
January 8 Tom Rodewald (Georgia Tech)
Title: Classifying Legendrian Cable Links of Negative Twist Knots
Abstract: Dalton, Etnyre, and Traynor classified Legendrian cable links when the companion knot is both uniformly thick and Legendrian simple, and Etnyre, Min, and Chakraborty classified all cable knots of uniformly thick knots. Using convex surfaces, we build on these results to classify cable links of negative twist knots in $(R^3, \xi_{std})$ – which are uniformly thick, but not Legendrian simple. In particular, we will show that there are pairs of links that are componentwise isotopic, but not isotopic as links. These are the first examples showing this behavior. This is joint work with Chatterjee, Etnyre, and Min.
December 25 Jun Zhang (USTC)
Title: Spectral capacities of submanifolds
Abstract: In this talk, we will discuss how to measure the size of submanifolds in a symplectic manifold. This is based on symplectic invariants that are constructed from spectra extracted from (Hamiltonian) Floer theory. Moreover, these invariants are concrete examples of spectral capacities. We will use these capacities to demonstrate the fundamental differences between Lagrangian submanifolds and symplectic submanifolds (or more generally nowhere coisotropic submanifolds). Meanwhile, we will also prove a quantitative Lagrangian control estimate that intriguingly relates these invariants. This talk is based on joint work with Dylan Cant.
December 18 Siyang Liu (University of Southern California)
Title: Symplectic Aspects of Representation Theory of Hypertoric Varieties
Abstract: In this talk we will discuss a symplectic model for T-equivariant hypertoric category \mathscr{O} as constructed by Braden-Licata-Proudfoot-Webster, which is the Fukaya-Seidel category of complexified complement of hyperplane arrangements associated to the corresponding hypertoric variety. This would provide a categorification of K-theoretic stable envelopes for hypertoric varieties. This is partly based on the recent work collaborated with S. Lee, Y. Li, and C. Y. Mak and the work in preparation with S. Ganatra, W. Li, and P. Zhou.
December 11 Bohan Fang (PKU) cancelled
Mirror symmetric Gamma conjecture for toric varieties and applications
I will describe how to compute oscillatory integrals in the LG mirror of a toric variety, producing the quantum cohomology central charge. In particular, I will explain two different methods: solving the GKZ equations and Fourier transforms. Such computation, combined with homological mirror symmetry, gives a proof of the Gamma II conjecture for toric Fanos. This work is based on the joint works with Peng Zhou, Konstantin Aleshkin and Junxiao Wang.
December 4 Yilin Wu (USTC)
Title: Relative Cluster categories and Higgs categories
Abstract: Cluster categories were introduced in 2006 by Buan-Marsh-Reineke-Reiten-Todorov in order to categorify acyclic cluster algebras without coefficients. Their construction was generalized by Amiot and Plamondon to arbitrary cluster algebras associated with quivers (2009 and 2011). A higher dimensional generalization is due to Guo (2011). Cluster algebras with coefficients are important since they appear in nature as coordinate algebras of varieties like Grassmannians, double Bruhat cells, unipotent cells,.... The work of Geiss-Leclerc-Schröer often yields Frobenius exact categories which allow to categorify such cluster algebras. In this talk, we will present the construction of the Higgs category (generalizing GLS'Frobenius categories E) and of the relative cluster category (generalizing the derived category of E) by using Ginzburg morphism which carries a canonical relative left 3-Calabi-Yau structure.
November 27 Junxiao Wang (PKU)
Title: Mirror Symmetric Gamma Conjecture for Toric GIT quotients via Fourier Transform
Abstract: The mirror symmetric Gamma conjecture states the equality between the central charges of a pair of mirror objects under homological mirror symmetry. In this talk, I will show how the mirror-symmetric Gamma conjecture for a toric Fano orbifold and its Landau-Ginzburg mirror arises as the Fourier transform of the equivariant version of mirror symmetric Gamma conjecture on C^r. This is joint work in progress with Konstantin Aleshkin and Bohan Fang.
Nov-20
Chris Brav (SIMIS)
Title: The cyclic Deligne conjecture and Calabi-Yau structures
Abstract:
The Deligne conjecture, many times a theorem, states that for a dg category C, the dg endomorphisms End(Id_C) of the identity functor-- that is, the Hochschild cochains-- carries a natural structure of 2-algebra. When C is endowed with a Calabi-Yau structure, then Hochschild cochains and Hochschild chains are identified up to a shift, and we may transport the circle action from Hochschild chains onto Hochschild cochains. The cyclic Deligne conjecture states that the 2-algebra structure and the circle action together give a framed 2-algebra structure on Hochschild cochains. We establish the cyclic Deligne conjecture, as well as a variation that works for relative Calabi-Yau structures on dg functors D --> C, more generally for functors between stable infinity categories. We discuss examples coming from oriented manifolds with boundary, Fano varieties with anticanonical divisor, and doubled quivers with preprojective relation. This is joint work with Nick Rozenblyum.
Oct 16 Ben Zhou (Tsinghua)
Title: Higher genus Gromov-Witten correspondences for log Calabi-Yau surfaces
Abstract: Strominger, Yau, and Zaslow (SYZ) phrased mirror symmetry as a duality between special Lagrangian fibrations over an affine manifold base. The Gross-Siebert program seeks to translate the SYZ conjecture into the language of algebraic geometry using toric degenerations and tropical geometry. From a toric log Calabi-Yau surface X with a smooth anticanonical divisor, one can construct a scattering diagram (which locally one associates a Poisson algebra) and its quantization using the Gross-Siebert program. One can then infer from the scattering diagram various kinds of Gromov-Witten invariants. I will explain the above terms, and how higher-genus correspondences between certain open, closed, and logarithmic Gromov-Witten invariants associated to the log Calabi-Yau surface X can be derived. Part of this is joint work with Tim Gr\"afnitz, Helge Ruddat, and Eric Zaslow.
Oct 9 Kenji Fukaya (Tsinghua)
Title: Categorification of Atiyah-Floer conjecture
Abstract: Atiyah-Floer conjecture relates two versions of Floer homologies one in gauge theory the other in symplectic geometry. I will explain how we “upgrade”it to a functorial equivalence of 2-A-infinity categories. Also I will explain its 2-3 dimensional topological Field theory and Lagrangian correspondences.
