Schedule

Titles and Abstracts

Hanwool Bae (IMS Chinese University of Hong Kong)

Title:

Wrapped Floer cohomology of $T^*_e G$ and Floer cohomology of the diagonal of $G/T^{-} \times G/T$

Abstract:

Let $G$ be a compact simply-connected semisimple Lie group and let $T$ be a maximal torus subgroup of $G$. We will discuss about the $A_{\infty}$-functor associated to a Lagrangian correspondence from $T^*G$ to $G/T^{-} \times G/T$. In particular, we will give a sketch of the proof that the $A_{\infty}$-homomorphism from the wrapped Floer cohomology of $T^*_e G$ to the Floer cohomology of the diagonal of $G/T^{-} \times G/T$ induces a ring isomorphism after a proper localization. This gives a new approach to a theorem proposed by Dale Peterson, which says that the homology of the based loop space of $G$ and the quantum cohomology of $G/T$ are isomorphic as rings. This is based on a work in progress with Naichung Conan Leung.

Huai-Liang Chang (Hong Kong University of Science and Technology)

Title:

Feynman rule of Gromov Witten theory

Abstract:

For compact Calabi Yau threefold, BCOV (1993) predicted Feynman rule that determine higher genus Gromov Witten invariants. Recently the conjecture is proved by the discovery of mixed spin p (msp) fields, using large N method. A consequence is that Fg is analytic. I shall introduce the setup and ideas.

Ryohei Chihara (University of Tokyo)

Title:

SO(3)-invariant G_2-manifolds

Abstract:

A G_2-manifold is a Ricci-flat Riemannian 7-manifold whose holonomy group is contained in the Lie group G_2. It is characterized by a closed and co-closed non-degenerate 3-form similar to a symplectic form. In this talk, we discuss SO(3)-invariant G_2-manifolds and their reduction together with related topics.

Ko Honda (UCLA)

Title:

Convex hypersurface theory in higher-dimensional contact topology

Abstract:

Convex surface theory and bypasses are extremely powerful tools for analyzing contact 3-manifolds. In particular they have been successfully applied to many classification problems. After reviewing convex surface theory in dimension three, we explain how to generalize many of their properties to higher dimensions. This is joint work with Yang Huang.

Hansol Hong (Yonsei University)

Title:

Bulk-deformed LG mirror for toric Fano surfaces

Abstract:

I will present an inductive algorithm to compute the bulk-deformed potentials for toric Fano surfaces via wall-crossing techniques and a tropical-holomorphic correspondence for holomorphic discs. As an application, we find an explicit relation between the oscillatory integrals of the bulk-deformed potentials and log descendant Gromov-Witten invariants, which recovers the previous result of Gross for P^2. This is a joint work with Yu-Shen Lin and Jingyu Zhao.

Hiroshi Iriyeh (Ibaraki University)

Title:

A proof of Mahler's conjecture for the volume product of three dimensional convex bodies

Abstract:

Mahler's conjecture is one of the classical open problems in the area of convex geometry. It states that for a centrally symmetric convex body K in the n-dimensional Euclidean space, the product of the volume of K and that of the polar body is greater than or equal to 4^n/n!. By a recent work by Artstein-Avidan, Karasev and Ostrover, this conjecture is closely related with Viterbo's isoperimetric-type conjecture for symplectic capacities. The two dimensional case of Mahler's conjecture was solved by Mahler in 1939. In this talk, we give a sketch of the proof of the three dimensional case. The conjecture is still open for n > 3. The talk is based on a joint work with Masataka Shibata.

Sangjin Lee (IBS Center for Geometry and Physics)

Title:

A higher-dimensional generalization of pseudo-Anosov surface automorphisms

Abstract:

In 80's, Thurston classified the mapping class group of orientable surfaces. A generic element of the mapping class group is of the pseudo-Anosov type. In 2014, from pseudo-Anosov surface automorphisms, Dimitrov, Haiden, Katzarkov and Kontsevich constructed Bridgeland stability conditions on the Fukaya category of the surface. They also gave a question asking the existence of higher-dimensional generalization of pseudo-Anosov automorphisms on symplectic manifolds. To answer their question, we found a construction of symplectomorphisms which preserve a stable Lagrangian lamination. In this talk, we will discuss the construction and some following questions.

Hsuan-Yi Liao (KIAS)

Title:

Formality theorem for dg manifolds

Abstract:

In late 90’s, Kontsevich proved a formality theorem which solved the major problem in deformation quantization and led to many new developments in mathematics. Inspired by Kontsevich's formulas and Shoikhet's conjecture, we establish a formality theorem for differential graded manifolds (a.k.a. Q-manifolds) which are a useful geometric notion unifying many important structures such as curved L-infinity algebras, derived intersections, complex manifolds and regular foliations. As an application, we prove a Duflo-type theorem for finite-dimensional dg manifolds. The talk is mainly based on joint works with Mathieu Stienon and Ping Xu.

Yat-Hin Suen (IBS Center for Geometry and Physics)

Title:

Reconstruction of TP2 via tropical Lagrangian multi-section

Abstract:

In this talk, I am going to talk about the reconstruction problem of the holomorphic tangent bundle TP2 of the complex projective plane. I will introduce the notion of tropical Lagrangian multi-section and cook up one by using the Fubini-Study metric. Then I will perform the reconstruction from this tropical Lagrangian multi-section. Walling-crossing phenomenon will occur in the reconstruction process.

Yoshihiro Sugimoto (NCTS)

Title:

Denseness of non-autonomous Hamiltonian diffeomorphisms

Abstract:

The Hamiltonian diffeomorphism group Ham(M,\omega) is the set of time 1 flows of time-dependent Hamiltonian vector fields. Ham(M,\omega) contains "autonomous" subset Aut(M,\omega) whose elements are time 1 flows of autonomous (=time-independent) Hamiltonian vector fields. One might expect that Aut(M,\omega) is a very small subset of Ham(M,\omega). In this talk, I will explain the denseness of the complement of Aut(M,\omega).

Ryosuke Takahashi (National Cheng Kung University)

Title:

Recent Developments in Z/2-harmonic spinors

Abstract:

Z/2-harmonic spinors are the codimension two subsets of a manifold which is determined by the zero loci of non-extendable harmonic spinors. It was initially a byproduct of the generalized Uhlenbeck's compactness theorem. Later on, the structure of Z/2-harmonic spinors appeared in the proof of compactness theorem for many gauge-theoretic equations such as G2-instantons, Kapustin-Witten equation, and Vafa-Witten equation. In this talk, we will review the recent development of Z/2-harmonic spinors and explain the difficulty we faced so far.

Hsian-Hua Tseng (Ohio State University)

Title:

descendant Hilb/Sym correspondence for the plane

Abstract:

Let S be a nonsingular surface. A version of the crepant resolution conjecture predicts that the descendant Gromov-Witten theory of Hilb^n(S), the Hilbert scheme of n points on S, is equivalent to the descendant Gromov-Witten theory of Sym^n(S), the n-fold symmetric product of S. In this talk we discuss how this works when S is ℂ^2. We explicitly identify a symplectic transformation equating the two descendant Gromov-Witten theories. We also establish a relationship between this symplectic transformation and the Fourier-Mukai transformation which identifies the (torus-equivariant) K-groups of Hilb^n(ℂ^2) and Sym^n(ℂ^2). This is based on joint work with R. Pandharipande.

Bai-Ling Wang (Australian National University)

Title:

K-theoretial virtual fundamental classes

Abstract:

I will reprot joint work with Bohui Chen and Jianxun Hu on K-theoretical Gromov-Witten and Hamiltonian Gromov-Witten invariants. The key ingredient in the definition of these invariants is the construction of K-theoretial virtual fundamental classes. As an application, we consider a Hamiltonian $G$-manifold $X$, and establish a K-theoretical relation between the Gromov-Witten invariants and the $L^2$-Hamiltonian Gromov-Witten invariants for the symplectic quotient orbifold $[X//G]$.

Hao Wen (Yau Mathematical Sciences Center)

Title:

Landau-Ginzburg model via L^2 Hodge theory

Abstract:

Let X be a non-compact Calabi-Yau manifold and f be a holomorphic function on X such that the critical set of f is compact. Under the assumption that f satisfies some strongly elliptic condition, I will describe an L^2 theoretic approach towards the deformation theory of the pair (X,f), including the Hodge to de Rham degeneration property and the construction of the corresponding Frobenius manifold structure.

Hikaru Yamamoto (Tokyo University of Science)

Title:

An epsilon regularity theorem for line bundle mean curvature flow

Abstract:

The line bundle mean curvature flow was defined by A. Jacob and S.-T. Yau to obtain deformed Hermitian Yang-Mills metrics on a line bundle over a Kahler manifold. In the context of mirror symmetry, a deformed Hermitian Yang-Mills metric in A-side corresponds to a special Lagrangian submanifold in B-side. In this talk, I would like to introduce an epsilon regularity theorem for the line bundle mean curvature flow and a general framework to get an epsilon regularity theorem. To explain the outline of the proof, I would like to introduce a scale-invariant monotone quantity, a notion of self-shrinkers and the Liouville type theorem for self-shrinkers. This is joint work with X. Han at Tsinghua University.

Hung-Yu Yeh (Academia Sinica)

Title:

Stability Filtrations, Weakly Ample Sequences and Numerical Vectors in Categories

Abstract:

Stability, first introduced by Mumford in the 1960's, is used as a tool to construct moduli space of sheaves on algebraic varieties and later generalized to objects in arbitrary abelian category. On the other hand, motivated by homological mirror symmetry conjecture and Douglas' Pi stability on the category of B-branes Bridgeland introduces stability conditions on triangulated categories which depends on the existence of Harder-Narasimhan (HN) filtration and central charges on the relevant K group of associated triangulated categories. In this talk, I would like to present main ideas in my current work and introduce a notion of stability filtration in arbitrary categories which is equivalent to the existence of HN filtration on objects. Indeed it is equivalent to existences of a zero morphism, a partial order on objects, and a collection of some universal sequences. One may give a suitable addition with this zero morphism on the Hom space which makes this category an additive category. Then with weakly ample sequences in the additive category embedded in an ambient triangulated category under suitable conditions, we could obtain a numerical polynomial or central charge of objects by calculating the Euler characteristic of weakly ample sequences and objects, inducing a partial order and HN filtration. At the end, I would give some easy examples in algebraic curves and surfaces.