Timetable

Abstract. We prove that there are at least as many exact embedded Lagrangian fillings as seeds for Legendrian links of finite type $\mathsf{ADE}$ or affine type $\widetilde{\mathsf{D}}\widetilde{\mathsf{E}}$. We also provide as many Lagrangian fillings with rotational symmetry as seeds of type $\mathsf{B}$, $\mathsf{G}_2$, $\widetilde{\mathsf{G}}_2$, $\widetilde{\mathsf{B}}$, or $\widetilde{\mathsf{C}}_2$, and with conjugation symmetry as seeds of type $\mathsf{F}_4$, $\mathsf{C}$, $\mathsf{E}^{(2)}_6$, $\widetilde{\mathsf{F}}_4$, or $\mathsf{A}^{(2)}_5$. These families are the first known Legendrian links with (infinitely many) exact Lagrangian fillings (with symmetry) that exhaust all seeds in the corresponding cluster structures beyond type $\mathsf{AD}$. This is a joint work with Youngjin Bae(INU) and Eunjeong Lee(IBS-CGP).

Abstract. Legendrian graphs naturally appear in the study of Weinstein manifolds with a singular Lagrangian skeleton, and a tangle decomposition of Legendrian links. I will introduce various invariant of Legendrian graphs including DGA, polynomial, moduli of sheaves, and their relationship. This is based on a joint project with Byunghee An, and partially with Tamás Kálmán and Tao Su.

Abstract. Holomorphic discs with geodesic lines between marked points are called popsicles. We define a compactification of popsicle holomorphic discs with interior insertions. This turns out to be rather delicate due to sphere bubbles with induced popsicle lines. We discuss various applications including Berglund-Hubsch homological mirror symmetry. This is based on several joint works with Dongwook Choa, Wonbo Jung and Hanwool Bae.

Abstract. This is a survey talk on mirror symmetry of Fano manifolds in view of cluster algebras. For a given smooth Fano variety X, a (week) Landau-Ginzburg mirror of X is a Laurent polynomial defined on a complex torus and it is known to be not unique in general. It has been expected that two mirrors of a same manifold are related to each other by so-called a mutation. In this talk, I will explain the mirror symmetry picture in terms of cluster algebras and introduce some related conjectures.

Abstract. I will review two possible approaches to obtain a mirror of a symplectic manifold in some mild setting. In either approaches, we will see that the key to understand mirror symmetry is Floer theory of Lagrangian submanifolds.

Abstract. Cone structures specify distinguished directions on complex manifolds. We review some geometric notions on cone structures and explain a recent joint work with Qifeng Li that if a 1-flat cone structure whose fibers are highest weight varieties comes from minimal rational curves, then it is locally symmetric.

Abstract. Effective, nef and semi-ample cones of minimal surfaces of general type with p_g = 0 will be discussed. Then I will provide examples of minimal surfaces of general type with p_g = 0 and 2 ≤ K^2 ≤ 9 which are Mori dream spaces. On these examples I also give explicit description of their effective cones with all negative curves. I will also provide non-minimal surfaces of general type with p_g = 0 that are not Mori dream surfaces. This is a joint work with Kyoung-Seog Lee.

Abstract. As a part of classification problems of Lagrangian submanifolds in a symplectic manifold, constructing monotone Lagrangian tori that are not Hamiltonian isotopic to each other is important. In this talk, I discuss how to construct infinitely many monotone Lagrangian tori and distinguish them in complete flag manifolds Fl(n) (n > 5). This talk is based on joint work in progress with Yunhyung Cho, Myungho Kim, Jae Hoon Kwon, and Euiyong Park.

Abstract. In joint work with Agustin Moreno, we propose a generalization of the Poincare-Birkhoff fixed point theorem. We start with a construction of global hypersurfaces of section in the spatial three-body problem, describe some return maps and suggest some generalizations of the Poincare-Birkhoff fixed point theorem. We use symplectic homology in the proof of our theorem

Abstract. Given a compact symplectic manifold and its mirror Landau-Ginzburg model, we expect that the quantum cohomology of the manifold is isomorphic to the Jacobian ring of the Landau-Ginzburg potential. Among many results concerning the expectation, we focus on Fukaya-Oh-Ohta-Ono’s geometric construction of the ring isomorphism in terms of a closed-open map. We will review the case when the target is not given by an embedded Lagrangian submanifold. Then we will see that the case becomes even more interesting when A- and B-models are equipped with group actions, by appearance of orbifold Jacobian algebras as closed B-models. Finally, a recent result on the construction of twisted Jacobian algebras in terms of matrix factorizations will be given, and we will see how it can lead us to the mirror symmetry problem of “stringy pairs”. A substantial part of the talk is baed on the joint work with Cheol-Hyun Cho.

Abstract. A rational blowdown surgery initially introduced by R. Fintushel and R. Stern and later generalized by J. Park is one of the simple but powerful techniques in the study of 4-manifolds topology. Note that a rational blowdown surgery replaces a certain linear chain of embedded 2-spheres by a rational homology 4-ball. In particular, a rational homology ball is a key ingredient in the construction of exotic smooth, symplectic 4-manifolds with small Euler characteristic and complex surfaces of general type with pg = 0. It also plays an important role in Q-Gorenstein smoothings and symplectic fillings of the link of complex surface singularities.

In this talk, first I’d like to briefly review what we have obtained in 4-manifolds topology by using a rational blowdown surgery. And then I’ll survey J. Evans and I. Smith’s recent result on rational homology balls embedded in CP2. This talk is based on their paper, Markov numbers and Lagrangian cell complexes in the complex projective plane, published in Geometry & Topology 22 (2018).