Program

For an isolated singularity, a variation operator maps relative homology to ordinary homology cycles in the Milnor fiber using a monodromy.  In this talk, we distinguish a particular element of Floer cohomology of a monodromy corresponding to the fundamental class of the boundary. We then define a symplectic analogue of the variation by an action of Floer cohomology of a monodromy. Lagrangian Floer cohomology of monodromy is also defined as the cohomology of the cone of the quantum cap action. This cohomology can be viewed as a categorificaton of the standard relation between Seifert form of the Milnor fiber and the variation operator. This is a joint work with Cheol-hyun Cho, Hanwool Bae, and Wonbo Jeong. 

In this talk we will introduce our recent work on blowup formulae of genus zero Gromov–Witten invariants. Via blowup along S × 0 of X × P1 we get a birational cobordism W between X and its blowup along S, denoted by \hat{X}. By taking r-th root of W along \hat{X} for sufficiently large r ≫ 1, via virtual localization technique and Pix- ton’s polynomiality of certain twisted Gromov–Witten invariants on r, when the normal bundle of S is positive (or nonnegative) in some sense, we prove some blowup formulae of genus zero Gromov–Witten invariants. As applications, our formulae implies that the symplectically rational connectedness is preserved by blowups along centers with positive (or nonnegative) normal bundles. If time permits, we will also mention the orbifold version of our model and the weighted-blowup formulae of genus zero orbifold Gromov–Witten invariants. This talk is based on joint work with Bohui Chen.

This talk is about my recent work with Camille Laurent-Gengoux. I will present our results about classifying singular foliations admitting a given leaf $L$ in a manifold $M$ and a given transverse model $(\mathbb{R}^d, \tau)$, where $\mathbb{R}^d$ is the fibre of a normal bundle of $L$ in $M$, and $\tau$ is a singular foliation in $\mathbb{R}^d$ admitting 0 as a leaf. Such a classification is motived by the fact that every foliation $\mathcal{F}$ induces a singular foliation in the fibres of a normal bundle, the \emph{transverse (singular) foliation}, and these transverse foliations at each point in $L$ are canonically isomorphic. These isomorphisms are given by the parallel transport of what one calls $\mathcal{F}$-connections.

The idea of this talk is to recover $\mathcal{F}$ given $(\mathbb{R}^d, \tau)$, and we will see that in a local neighbourhood around $L$ every foliation admitting $(\mathbb{R}^d, \tau)$ as transverse model is given by an associated connection of a curved Yang-Mills gauge theory, a generalised gauge theory I have developed last year. Usually, the horizontal distribution of a flat connection gives rise to a regular foliation, while our condition roughly says that the curvature is related to the field strength of a curved gauge theory. This is a natural enhancement, allowing singular foliations as a consequence, and this construction is naturally invariant of the choice of $\mathcal{F}$-connections.

Gamma conjecture II for quantum cohomology of a Fano manifold   was proposed by Galkin, Golyshev and Iritani. It expects to describe the asymptotic behavior of flat sections of the Dubrovin connection near the irregular singularity, in terms of a full exceptional collection of   and the Gamma-integral structure. We give a criterion for this conjecture, based on Dubrovin’s result of analytic continuation of semisimple Frobenius manifolds. As an application, we use this criterion to verify this conjecture for smooth quadric hypersurfaces. This talk is based on a joint work with Xiaowen Hu.

Abstract. In symplectic topology, there has been substantial interests in constructing monotone Lagrangian tori that are not related by any symplectomorphisms. In this talk, I discuss how to use cluster algebras, toric degenerations, representation theory for constructing infinitely many monotone Lagrangian tori and distinguishing them in complete flag manifolds of arbitrary type (except in some low-dimensional cases). This talk is based on a joint work with Yunhyung Cho, Myungho Kim, and Euiyong Park.

For a given contact manifold, it is a fundamental question in symplectic topology in how many ways the contact manifold can be written as the boundary of a symplectic filling. In this talk, we discuss uniqueness results on symplectic fillings using J-holomorphic curves, focusing on the case when the contact boundary is the standard unit cotangent bundles over spheres. We present applications to symplectic cobordisms and symplectic hyperplane sections. This talk is based on joint work with Takahiro Oba.

In this talk, we will show that the \partial\bar{\partial}-lemma holds on the non-Kahler CY threefolds constructed by Hashimoto and Sano (arXiv:1902.01027). As a consequence, they have Hodge decompositions and one can study their B-model via variation of Hodge structure.

It has been a continuing interest, often with profound importance, in understanding the geometric and topological relationship between a Hamiltonian G-space Y and its symplectic reduction X. In this talk, we provide precise relations between the equivariant Lagrangian Floer theory of a G-invariant Lagrangian in Y and the (ordinary) Lagrangian Floer theory of its reduced Lagrangian in X, which refines a conjecture of Teleman in his ICM address. An important ingredient of the proof is an equivariant extension of the (Lagrangian) Correspondence Tri-modules, introduced by Fukaya in absence of symmetry, as modules over equivariant Floer complexes, introduced by Yoosik Kim, Siu Cheong Lau and Xiao Zheng.

If time permits, we will discuss how the above fits into the framework of 3d Mirror Symmetry, which in particular associates Hamiltonian G-spaces (as branes over the Higgs branch pt/G) with holomorphic Lagrangian branes (as branes over the Coulomb branch BFM(G), the BFM space of G). Joint work with Prof. Nai Chung Conan Leung and Siu Cheong Lau.

Special Lagrangians are an important class of area minimizers introduced by Harvey-Lawson. Strominger-Yau-Zaslow conjecture predicts that Calabi-Yau manifolds admit special Lagrangian fibration and mirrors can be constructed via the dual torus fibration. Moreover, special Lagrangians are conjecturally to be the stable objects for Fukaya categories of Calabi-Yau manifolds. In this talk, I will explain some construction of certain special Lagrangian 3-spheres in Calabi-Yau 3-folds with K3-fibration and small fibres. In particular, these special Lagrangian spheres collapse to certain geodesics of the base of the K3-fibration when the volume of K3 fibres shrinks to zero. This is a joint work in progress with Shih-Kai Chiu. 

We start with the conormal bundle of a knot $K_0$ in $\mathbb{R}^3$. Suppose that it is isotoped in $T^*\mathbb{R}^3$ to a Lagrangian submanifold which intersects the zero section cleanly along another knot $K_1$. Then we ask whether the knot type of $K_1$ is different from that of $K_0$. (This is a "local" version of a question by Smith.) In this talk, we will see that the studies on knot contact homology (the Legendrian contact homology of the unit conormal of knots) by Ng can be applied to this problem. We then deduce some results that restricts the knot type of $K_1$ from that of $K_0$.

In the 1990s, Taubes showed the equivalence between the number of certain pseudo-holomorphic curves (Gromov invariants) and Seiberg-Witten invariants in four-dimensional symplectic manifolds. Embedded Contact Homology (ECH), introduced by M. Hutchings, is a three-dimensional Floer homology analog of Gromov invariants and is defined for closed three-dimensional contact manifolds. ECH has become a powerful tool in the study of three-dimensional Reeb flows. It is well-known to be isomorphic to Monopole Floer homology defined by Kronheimer and Mrowka. On the other hand, prior to the introduction of ECH, important research was conducted by Hofer, Wysocki, and Zehnder. They constructed disk-like Birkhoff sections on convex three-dimensional contact spheres  from pseudo-holomorphic curves. In this talk,  I will explain a relationship between the work of Hofer, Wysocki, Zehnder and ECH, and introduce the results obtained from such an approach. 

An example of Apostolovet al. indicate that the condition of K-stability may not be correct one for general polarised manifolds. Szekelyhidi modified definition of K-stability by filtration and stated a variant of the Yau-Tian-Donaldson conjecture. We will talk about our proof of this variant of YTD conjecture for toric manifolds and homogeneous toric bundles. This is jointed with Li An-Min and Lian Zhao.

Geometric quantization on symplectic manifolds plays an important role in representation theory and mathematical physics, deeply relating to symplectic geometry and differential geometry. A crucial problem is to understand the relationship among geometric quantizations associated to different polarizations.  In this talk, we will discuss  the construction of mixed polarization by T-symmetry on Kähler manifolds and the geometric description of the quantum space associated with this mixed polarization.  This is a  joint work with Naichung Conan Leung.

For a union of exact Lagrangians L, the Maruer-Cartan algebra of L is a source of interesting quiver algebras. In this talk we consider the case when L is the Lagrangian skeleton of a multiplicative hypertoric variety. The corresponding quiver algebra gives a noncommutative mirror to the multiplicative hypertoric variety. This talk is based on joint works with Lau and Ma.

The moduli space of spatial polygons is known as a symplectic manifold equipped with both Kahler and real polarizations. In this talk, we will construct morphisms of operads $\mathsf{f}_{Kah}$ and $\mathsf{f}_{re}$ by using the quantum Hilbert spaces $\mathcal{H}_{Kah}$ and $\mathcal{H}_{re}$ associated to the Kahler and real polarizations respectively. Moreover, we will relate the two morphisms $\mathsf{f}_{Kah}$ and $\mathsf{f}_{re}$, and prove the equality $\dim \mathcal{H}_{Kah} = \dim \mathcal{H}_{re}$ in general setting. This operadic framework is regarded as a development of the recurrence relation method by Kamiyama (2000) for proving $\dim \mathcal{H}_{Kah} = \dim \mathcal{H}_{re}$ in a special case.