We give a gentle introduction to Morse and Floer theories. Morse theory tells us that the critical points of a generic function on a smooth manifold determines topological information. Floer theory can be seen as an infinite-dimensional version of Morse theory, and it enables us to prove the Arnold conjecture. In this talk, we focus on its basic ideas as well as history.

We present a new construction of mirror pairs of Calabi-Yau manifolds by smoothing normal crossing varieties, consisting of two quasi-Fano manifolds. We introduce a notion of mirror pairs of quasi-Fano manifolds with anticanonical Calabi-Yau fibrations using conjectures about Landau-Ginzburg models. Utilizing this notion, we give pairs of normal crossing varieties and show that the pairs of smoothed Calabi-Yau manifolds satisfy the Hodge number relations of mirror symmetry. We consider quasi-Fano threefolds that are some blow-ups of Gorenstein toric Fano threefolds and build 6518 mirror pairs of Calabi-Yau threefolds, including 79 self-mirrors. 

For a reductive algebraic group G, a normal G-variety is called spherical if it contains an open orbit under the action of a Borel subgroup of G. The class of spherical varieties contains several important geometric objects which were studied independently, for example, toric varieties, flag varieties, horospherical varieties, group embeddings, and symmetric varieties. Interestingly, the normal equivariant embeddings of a given spherical homogeneous space are classified by combinatorial objects called colored fans, which generalize the fans appearing in the classification of toric varieties. As a result, many geometric properties of Fano spherical varieties can be described from the (algebraic) moment polytopes encoding the structure of representation of G on the spaces of sections of tensor powers of the anticanonical line bundle. For instance, we have a criterion for K-stability of a smooth Fano spherical variety in terms of the barycenter of its moment polytope with respect to the Duistermaat-Heckman measure and data associated to the corresponding spherical homogeneous space. In this talk, we discuss geometric properties of Fano spherical varieties and the use of their moment polytopes through various examples.

In 1980, Eisenbud introduced the notion of matrix factorizations to study free resolutions over a hypersurface ring. Roughly speaking, a matrix factorization of a polynomial f is a pair of matrices (A,B) where both AB and BA are f times the identity matrix. Matrix factorizations also play a significant role in singularity theory and Landau-Ginzburg theory, for instance, D-branes of type B can be obtained from matrix factorizations of the superpotential function. Another intersting application of matrix factorizations can be found in a study of so-called ACM vector bundles on hypersurfaces. ACM bundles are geometric analogues of maximal Cohen-Macaulay modules, and the existence can help to understand the algebro-geometric nature of the underlying space. In this talk, I will discuss construction of Ulrich bundles, ACM bundles with even stronger properties, on some cubic hypersurfaces as an application of matrix factorization. 

Burban and Drozd (2017) classified all Cohen-Macaulay modules over degenerate cusps. For the degenerate cusp defined by xyz, its mirror is given by a pair of pants (Abouzaid, Auroux, Efimov, Katzarkov and Orlov). Our goal is to find explicit objects in the Fukaya category of the pair of pants, which correspond to every Cohen-Macaulay modules in Burban and Drozd's list.

In this talk, we will particularly focus on the formulation of localized mirror functor under the language of deformation theory developed by Fukaya, Oh, Ohta and Ono. As its application, we find canonical forms of the matrix factors of xyz corresponding to each object in the Fukaya category. Then we show that they also match with the Cohen-Macaulay modules in Burban and Drozd's list.

This is a joint work with Cheol-Hyun Cho, Wonbo Jeong and Kyoungmo Kim.

 In this talk, we first give some basic facts on exceptional collections and describe what Kuznetsov’s conjecture is. After that, we provide all the exceptional collections of maximal length up to normalization and cyclic permutation and prove that they are in fact full.

In this talk, I would like to introduce some tools which allow us to study Symplectic topology in topological/combinatorial ways. The tools are Weinstein handle decompositions and Lefschetz fibrations. If time allows, I will introduce my recent joint work with Dongwook Choa and Dogancan Karabas, which constructs a pair of diffeomorphic manifolds having different symplectic structures, from Lefschetz fibrations. In the talk, no preliminary knowledge of symplectic topology will be required.  

In this talk, I will introduce the Total dual Variety of Minimal tangents and show its application on studying bigness of tangent bundle based on the joint work with Jeong-Seop Kim and Yongnam Lee