Symplectic Geometry in Gongju I

Title & Abstract

Speaker

Jungsoo Kang (Seoul National University)

Title

On the Clarke duality

Abstract

In this series of talks, I will explain the Clarke duality and its connection to symplectic homology. Some applications will also be discussed. 

Speaker

Myeonggi Kwon (Jeonbuk National University)

Title

Entropy of symplectic Torelli classes via Floer homology I

Abstract

In this talk, we are interested in a topological entropy of symplectomorphisms which act trivially on homology. A connected component of such symplectomorphisms is called a symplectic Torelli class, and the existence of a symplectic Torelli class with positive entropy is particularly interesting as it shows a difference of symplectic topology from smooth topology. We discuss for certain Liouville domains how to find a symplectic Torelli class with positive entropy using various constructions in Floer theory. This talk is based on joint work with Joontae Kim.

Speaker

Joontae Kim (Sogang University)

Title

Entropy of symplectic Torelli classes via Floer homology II

Abstract

Recently, Buhovsky--Tanny obtained a Sikorav-type energy bound for Floer trajectories passing to the contact boundary of a subdomain in a symplectic manifold. This is a key ingredient to exhibit a local-to-global phenomenon in Floer theory. We explain this idea applies to Floer trajectories for Floer homology of symplectomorphisms as well. This is joint work in progress with Myeonggi Kwon.

Speaker

Woongbae Park (Syracuse University)

Title

Harmonic maps, bubbling, and the conformal heat flow

Abstract

One of Gromov' prestigious results is the compactness theorem of J-holomorphic curves in symplectic manifolds. An analogous result for minimal surfaces is obtained by Sacks-Uhlenbeck. Under such settings, limits of weakly converging subsequences can be regarded as the bubbles attached to the body map. In this talk, we briefly go over basic ideas of bubbling in harmonic maps and their heat flow, and introduce an attempt to capture bubbles under some geometric flow, called conformal heat flow. Then we cover some results of conformal heat flow.

Speaker

Wonbo Jeong (Seoul National University)

Title

Introduction to the homological mirror symmetry about singularities

Abstract

In this talk, I will introduce a version of HMS about isolated singularities and their symmetry groups. It is a category equivalence between Fukaya category associated to given singularity and matrix factorization category of the mirror. A construction of such Fukaya category was not known and in the joint work with Choel-Hyun Cho and Dongwook Choa, we proposed the general construction. I will explain briefly that construction. It can be thought as a categorification of the new Floer cohomology (which will be introduced in the talk of Dr. Hanwool Bae). This talk is based on two joint works, one with Choel-Hyun Cho and Dongwook Choa and the other with Hanwool Bae, Cheol-Hyun Cho and Dongwook Choa.

Speaker

Hanwool Bae (Seoul National University)

Title

Floer theory for the variation operator of an isolated singularity

Abstract

For an isolated hypersurface singularity, there is a linear operator from the relative homology of its Milnor fiber to its absolute homology, called variation operator. We discuss its Floer theoretic enhancement, which is a functor from the wrapped Fukaya category of the Milnor fiber to its full subcategory consisting of proper objects. We further introduce a Floer theory, which categorifies the Seifert forms on Milnor fibers. The main ingredient of the construction is a certain cohomology class of the fixed point Floer cohomology of the inverse of the monodromy. This talk is based on joint work with Cheol-Hyun Cho, Dongwook Choa and Wonbo Jeong.

Speaker

Sungho Kim (Seoul National University)

Title

Rabinowitz Floer homology for prequantization bundles and Floer Gysin sequence

Abstract

In this talk, we introduce the Floer Gysin sequence, a Floer theoretic analogue of the classical Gysin sequence, which connects the Rabinowitz Floer homology of a prequantization bundle to the quantum homology of its base manifold. Similar to the classical Gysin sequence, the maps in the Floer Gysin sequence satisfy relations regarding the homologies’ product structures. A key idea of the arguments is the process of lifting holomorphic spheres to Floer cylinders and Floer pairs of pants, which will be briefly discussed. We will also explore computations and applications utilizing the Floer Gysin sequence. This talk is based on joint work with Joonghyun Bae and Jungsoo Kang.

Speaker

Wonjun Lee (Seoul National University)

Title

Invariance of Rabinowitz Floer Homology

Abstract

In 2020, Lazarev introduced a new class of contact manifolds known as asymptotically dynamically convex (ADC) contact manifolds and proved that the positive symplectic homology of a Liouville domain with ADC boundary is independent of the filling. In this talk, we improve Lazarev’s result to the filtered symplectic homology. Moreover, we show that the Rabinowitz Floer homology is independent of the filling. Then, we discuss a few applications. This talk is based on a joint work with Jungsoo Kang.