Time
Dec 28-30, 2024
Venue
C1124, the Material and Science Research Building (section C), East Campus of USTC
Organizer
River CHIANG (National Cheng Kung University)
Jun ZHANG (University of Science and Technology of China)
Secretary
Chun TAN, nanbei0104@ustc.edu.cn
Organizing Parties
Institute of Geometry and Physics, USTC
Speakers
Guanheng Chen (Shenzhen University)
River Chiang (National Cheng Kung University)
Sheng-Fu Chiu (National Tsing Hua University)
Huagui Duan (Nankai University)
Wenmin Gong (Beijing Normal University)
Sungho Kim (Institute of Geometry and Physics, USTC)
Tsung-Ju Lee (National Cheng Kung University)
Wenyuan Li (University of Southern California)
Qing Liu (Nankai University)
Yu Pan (Tianjin University)
Xiudi Tang (Beijing Institute of Technology)
Weiwei Wu (Zhejiang University)
Yuan Yao (University of Nantes)
Tianyu Yuan (Eastern Institute of Technology, Ningbo)
Zhengyi Zhou (Academy of Mathematics and Systems Science, CAS)
Schedule (each talk 50 mins)
Saturday morning (12/28):
09:00 - 09:50 Yu Pan
10:00 - 10:50 Qing Liu
11:10 - 12:00 Tsung-Ju Lee
Saturday afternoon (12/28):
14:00 - 14:50 Sheng-Fu Chiu
15:00 - 15:50 Wenyuan Li
16:10 - 17:00 River Chiang
Sunday morning (12/29):
09:00 - 09:50 Yuan Yao
10:00 - 10:50 Huagui Duan
11:10 - 12:00 Zhengyi Zhou
Sunday afternoon (12/29):
14:00 - 14:50 Weiwei Wu
15:00 - 15:50 Sungho Kim
16:10 - 17:00 Wenmin Gong
Monday morning (12/30):
09:00 - 09:50 Xiudi Tang
10:00 - 10:50 Guanheng Chen
11:10 - 12:00 Tianyu Yuan
Titles & Abstracts
Guanheng Chen (Shenzhen University)
Title: Comparing PFH and HF spectral invariants
Abstract: Periodic Floer homology and quantitative Heegaard Floer homology are respectively Floer type invariants for symplectomorphisms on surfaces. There are two types of numerical invariants emerge respectively from these two Floer theories called PFH spectral invariants and link spectral invariants. In my talk, I will present results on the relation between these two kinds of spectral invariants.
River Chiang (National Cheng Kung University)
Title: TBA
Abstract: TBA
Sheng-Fu Chiu (National Tsing Hua University)
Title: Coisotropic set and coisotropic rigidity
Abstract: I will apply microlocal sheaf theory to give an alternative definition of coisotropic sets and show that this new notion underlies a hierarchy of coisotropic properties ranging from classical coisotrpicity, to Kashwara-Schapira's cone coisotropicity, to Guillermou-Viterbo’s gamma coisotropicity. Moreover, our notion also improves the famous C0 rigidity of smooth coisotropic set. This is a joint work in preparation with Yuichi Ike and Tomohiro Asano.
Huagui Duan (Nankai University)
Title: Maslov-type index theory and closed orbits
Abstract: In this talk, I will introduce two kinds of closed orbit problems, i.e., closed geodesics on manifolds and closed orbits on hypersurfaces with the fixed energy. Then I will introduce some recent progress in this field, and explain how to deal with these problems by using Maslov-type index theory.
Wenmin Gong (Beijing Normal University)
Title: The unbounded Lagrangian spectral norm and wrapped Floer cohomology
Abstract: In this talk, I will investigate the question of whether the spectral metric on the orbit space of a fiber in the disk cotangent bundle of a closed manifold, under the action of the compactly supported Hamiltonian diffeomorphism group, is bounded. We show that the spectral metric on the orbit space of an admissible Lagrangian is bounded if and only if the wrapped Floer cohomology vanishes. Consequently, the Lagrangian Hofer diameter of the orbit space for any fiber in the disk cotangent bundle of a closed manifold is infinite.
Sungho Kim (Institute of Geometry and Physics, USTC)
Title: Rabinowitz Floer homology of prequantization bundles
Abstract: We will review the foundational aspects of Rabinowitz Floer Homology (RFH), exploring its properties, computational methods, and applications. A key focus will be the computation of RFH for prequantization bundles via Floer Gysin sequence, illustrated with explicit examples. Additionally, we will briefly discuss the Lagrangian and relative versions of RFH. This is based on joint works with Hanwool Bae, Joonghyun Bae, Jun Zhang, and Jungsoo Kang.
Tsung-Ju Lee (National Cheng Kung University)
Title: Mirror duality between singular Calabi—Yau varieties: the trivial nef-partition case
Abstract: In this talk, I will explain the ideas as well as our ongoing program on mirror symmetry for singular Calabi—Yau varieties; in short, for a given singular CY varieties arising from double covers of toric manifolds branching over a nef-partition, we can construct another singular CY variety such that the period integrals of one compute the orbifold GW invariants of the other. I will especially explain the proof when the nef-partition is trivial. This is based on joint works with Shinobu Hosono, Bong Lian, and Shing-Tung Yau.
Wenyuan Li (University of Southern California)
Title: Lagrangian cobordisms and microlocal sheaf theory
Abstract: In symplectic geometry, Lagrangian cobordism is an important relation between Lagrangian submanifolds introduced by Arnol’d. Later, Biran and Cornea noticed the relationship between Lagrangian cobordisms and Fukaya categories. We will show the relationship between Lagrangian cobordisms and another closely related invariant coming from microlocal theory of sheaves. We explain how Lagrangian cobordisms between submanifolds in cotangent bundle define sheaves on the base manifold, how they induce equivalences and iterated cone decompositions in the Tamarkin category, and how the persistence-like distance is related to the shadow distance of Lagrangian cobordisms, which leads to Lagrangian intersection estimations. This is joint work with Tomohiro Asano and Yuichi Ike.
Qing Liu (Nankai University)
Title: Beyond Hyperbolicity: Morse boundaries
Abstract: The Morse boundary of a proper geodesic metric space is a generalization of the Gromov boundary of a hyperbolic space. It is a quasi-isometry invariant to study "hyperbolic directions" in the space. That is, a quasi-isometry between two spaces induces a homeomorphism on their Morse boundaries. This homeomorphism satisfies a variety of metric properties including bi-hölder, quasi-conformal, quasi-möbius and power quasisymmetric. In this talk, we will give a brief introduction about Morse boundaries and investigate these structures on the Morse boundary which determine the interior space up to a quasi-isometry.
Yu Pan (Tianjin University)
Title: Augmentation variety and Exact Lagrangian fillings
Abstract: Exact Lagrangian surfaces are important objects in the derived Fukaya category. Augmentations are objects of the augmentation category, which is the contact analog of the Fukaya category. In this talk, we discuss various relations between augmentations and exact Lagrangian surfaces. We also use immersed Lagrangian cobordisms to understand the cluster structure of augmentation variety.
Xiudi Tang (Beijing Institute of Technology)
Title: Symplectic classification of compact almost-toric systems of dimension four
Abstract: Almost-toric systems are important in mirror symmetry. We give a classification of 4-dimensional compact almost-toric systems up to fiber-preserving symplecctomorphisms. This generalizes the classification by Pelayo--Vu Ngoc on simple semitoric systems and that by the speaker together with Pelayo and Palmer on semitoric systems, both in dimension four. The extra difficulty for almost-toric systems is the lack of a global circle action. The polygon invariant is replaced by an almost-toric closed disk, and we give appropriate notions of focus-focus label and twisting indices in the almost-toric case.
Weiwei Wu (Zhejiang University)
Title: Nontriviality of loops of ball-embeddings and camel obstructions in closed four manifolds
Abstract: Symplectic Camel theorem observed by Elisashberg and Gromov represents a classical rigidity phenomenon in symplectic geometry. Classically, the symplectic camel problem considers the connectedness of two standard ball-embeddings in an open subset of \mathbb{R}^{2n}. In this talk, we explain how we use this idea to find a nontrivial loop in the space of ball-embeddings of a compact symplectic four manifold. This method has good potentials to generalize our result to a vast class of compact symplectic manifolds. This is a joint work with Zhengyi Zhou and Yi Du.
Yuan Yao (University of Nantes)
Title: anchored symplectic embeddings
Abstract: Given two four-dimensional symplectic manifolds, together with knots in their boundaries, we define an "anchored symplectic embedding" to be a symplectic embedding, together with a two-dimensional symplectic cobordism between the knots (in the four-dimensional cobordism determined by the embedding). We use techniques from embedded contact homology to determine quantitative critera for when anchored symplectic embeddings exist, for many examples of toric domains. In particular we find examples where ordinarily symplectic embeddings exist, but they cannot be upgraded to anchored symplectic embeddings unless one enlarges the target domain. This is joint work with Michael Hutchings, Agniva Roy and Morgan Weiler.
Tianyu Yuan (Eastern Institute of Technology, Ningbo)
Title: A Morse A infinity-model for the higher-dimensional Heegaard Floer homology of cotangent fibers
Abstract: Given a smooth manifold and tuples of basepoints, we define a Morse-type A infinity-algebra, called the based multiloop algebra, as a graded generalization of the braid skein algebra due to Morton-Samuelson. For example, when the braid skein algebra is the Type A double affine Hecke algebra (DAHA). The A infinity-operations couple Morse gradient trees on a based loop space with Chas-Sullivan type string operations. We show that, after a certain base change, it is equivalent to the wrapped higher-dimensional Heegaard Floer A infinity-algebra of disjoint cotangent fibers. This is joint work with Ko Honda, Roma Krutowski, and Yin Tian.
Zhengyi Zhou (Academy of Mathematics and Systems Science, CAS)
Title: Unknottedness of symplectic submanifold fillings
Abstract: We will review Siefring’s intersection theory for higher dimensional symplectic manifolds. Then we will show that any symplectic filling of the standard contact submanifold (S^{2n-1},\xi_{\std}) of (S^{2n+1},\xi_{\std}) in (D^{2n+2},\omega_{\std}) is smoothly unknotted.
