2024几何与拓扑会议
2024 Geometry and Topology Conference
2024几何与拓扑会议
2024 Geometry and Topology Conference
June 2^{th} - June 7^{th}, Shanghai, China
June 2^{th} - June 7^{th}, Shanghai, China
Organizing Committee
Aleksander Doan (University College London, UK.)
Song Sun (Zhejiang University and UC Berkeley)
Shi Wang (ShanghaiTech University, China)
Ruobing Zhang (Princeton University, USA.)
Sponsor
Institute of Mathematical Sciences, ShanghaiTech University
The Organizing Committee of 2024 Geometry & Topology Conference
Contact
Shi Wang (wangshi@shanghaitech.edu.cn)
Yijie Lu (luyj2@shanghaitech.edu.cn)
The Organizing Committee of 2024 Geometry & Topology Conference
8：30-9：00 | Opening Ceremony |
9：30-10：30 | Wilderich Tuschmann (Karlsruhe Institute of Technology, DE) Title:Spaces and moduli spaces of (Riemannian) metrics |
10：30-11：00 | Break |
11：00-12：00 | Qiongling Li (Nankai University, CN) Title: Index and total curvature of minimal surfaces in noncompact symmetric spaces and wild harmonic bundles |
12:00-14:00 | Noon break |
14:00-15:00 | Siarhei Finski (École Polytechnique, FR) Title: On the geometry at infinity on the space of Kähler potentials |
15:00-15:30 | Break |
15:30-16:30 | Yaxiong Liu (University of Maryland, USA) Title: The eigenvalue problem of complex Hessian operators |
Title & Abstract
JUNE 03 (MONDAY)
08.30-09.00 Opening Ceremony
09.30-10.30 Wilderich Tuschmann (Karlsruhe Institute of Technology, DE)
Title:Spaces and moduli spaces of (Riemannian) metrics
Abstract: Consider a smooth manifold with a Riemannian metric satisfying some sort of geometric constraint like, for example, positive scalar curvature, non-negative Ricci or negative sectional curvature, being Einstein, Kähler, Sasakian, of special holonomy, etc.
A natural question to ponder is then what the space of all such metrics does look like.
Moreover, one can also study this question for the corresponding moduli spaces of metrics,i.e., quotients of the former by the diffeomorphism group of the manifold, acting by pulling back metrics. These spaces are customarily equipped with the topology of smooth convergence on compact subsets and the quotient topology, respectively, and their topological properties then provide the right means to measure 'how many' different metrics and geometries the given manifold actually does exhibit. The history of the subject as a whole indeed goes back more than a century, and since H. Weyl’s early result on the connectedness of the space of positive Gaussian curvature metrics on the two-sphere and the foundings of Teichmüller, infinite- dimensional manifold and Lie group theory, uniformization and geometrization, the study of spaces of metrics and their corresponding moduli has been a topic of interest for differential geometers, global and geometric analysts and topologists alike.
In my talk, I will provide a gentle introduction to the subject with a focus on lower curvature bounds and present recent results and open questions about the global topological properties of moduli spaces of nonnegatively curved Riemannian metrics on manifolds, and as well, if time permits, also corresponding facts and questions for the moduli of nonnegative curvature metrics on RCD spaces.
11.00-12.00 Qiongling Li (Nankai University, CN)
Title: Index and total curvature of minimal surfaces in noncompact symmetric spaces and wild harmonic bundles
Abstract: We prove two main theorems about equivariant minimal surfaces in arbitrary nonpositively curved symmetric spaces extending classical results on minimal surfaces in Euclidean space. First, we show that a complete equivariant branched immersed minimal surface in a nonpositively curved symmetric space of finite total curvature must be of finite Morse index. It is a generalization of the theorem by Fischer-Colbrie, Gulliver-Lawson, and Nayatani for complete minimal surfaces in Euclidean space. Secondly, we show that a complete equivariant minimal surface in a nonpositively curved symmetric space is of finite total curvature if and only if it arises from a wild harmonic bundle over a compact Riemann surface with finite punctures. Moreover, we deduce the Jorge-Meeks type formula of the total curvature and show it is an integer multiple of $2pi/N$ for $N$ only depending on the symmetric space. It is a generalization of the theorem by Chern-Osserman for complete minimal surfaces in Euclidean n-space. This is joint work with Takuro Mochizuki (RIMS).
14.00-15.00 Siarhei Finski (École Polytechnique, FR)
Title: On the geometry at infinity on the space of Kähler potentials
Abstract: For a complex projective manifold polarised by an ample line bundle, we study the geometry at infinity on the space of all positive metrics on the line bundle. We show that this geometry is related to some asymptotic properties of submultiplicative filtrations on the section ring. This establishes a certain metric relation between differential geometry — in the form of the space of Kähler metrics and geodesic rays in it — and algebraic geometry — in the form of test configurations and associated filtrations.
15.30-16.30 Yaxiong Liu (University of Maryland, USA)
Title: The eigenvalue problem of complex Hessian operators
Abstract: In a very recent pair of nice papers of Badiane and Zeriahi, they consider the eigenvalue problem of complex Monge-Ampere and complex Hessian, and show that the C^{1,ar{1}}-regularity of eigenfunction for MA and C^alpha-regularity for complex Hessian. They posed a question about the C^{1,1}-regularity of the eigenfunction and the uniqueness. We give a positive answer and show the C^{1,1}-regularity and uniqueness of the eigenfunction. We also derive a number of applications, including a bifurcation-type theorem and geometric bounds for the eigenvalue. This is a joint work with Jianchun Chu and Nicholas McCleerey.
JUNE 04 (TUESDAY)
9.30-10.30 Shouhei Honda (The University of Tokyo, JP)
Title: Almost rigidity to flat torus via harmonic map
Abstract: In this talk we provide a couple of rigidity results on flat tori in terms of harmonic maps. This is a joint work with Christian Ketterer, Ilaria Mondello, Raquel Perales and Chiara Rigoni.
11.00-12.00 Junsheng Zhang (University of California, Berkeley, USA)
Title: On Calabi-Yau metrics asymptotic to cones
Abstract: We proved a ‘’no semistability at infinity" result for complete Calabi-Yau metrics asymptotic to cones, by eliminating the possible appearance of an intermediate K-semistable cone in Donaldson-Sun's 2-step degeneration theory. As a consequence, we establish a polynomial convergence rate result and a classification result for complete Calabi-Yau manifolds with Euclidean volume growth and quadratic curvature decay. This is based on joint work with Song Sun.
14.00-15.00 Beibei Liu (The Ohio State University, USA)
Title: Uniform spectral gap and orthogeodesic counting for Kleinian groups
Abstract: Strongly convergent sequences of hyperbolic manifolds arise naturally in the study of Kleinian group representations, for example, the Dehn surgeries on hyperbolic knots. It turns out that such sequences usually have uniform control on the geometry and dynamics, such as the uniform convergence of small eigenvalues of the Laplacian, and the Patterson-Sullivan measures. We will talk about the uniform convergence results in this talk and apply them to count uniformly along the sequence the number of simple closed geodesics and orthogeodesics. This is joint work with Franco Vargas Pallete.
15.30-16.30 Xiaolei Wu (Fudan University, CN)
Title: On the homology of big mapping class groups
Abstract: I will start slowly with the basic notions on homology of groups. Then I will give a review on the calculations of homology of mapping class groups for finite type surfaces. After that I will discuss what is the story in the case of infinite type surfaces. In particular, I will discuss how one can calculate the homology of mapping class groups of some well-know surfaces, including disk minus Cantor set. This is based on joint works with Martin Palmer.
JUNE 05 (WEDNESDAY)
09.30-10.30 Xujia Chen (Harvard University, USA)
Title: A product operation on disk fiber bundles, and a configuration space with mouse diagrams
Abstract: In this talk we will be concerned with smooth, framed fiber bundles whose fibers are the standard d-dimensional disk, trivialized along the boundary. "Kontsevich's characteristic classes" are invariants defined for these bundles: given such a bundle pi:E o B, we can associate to it a collection of cohomology classes in H^*(B). On the other hand, there is a "bracket operation" for these bundles defined by Sander Kupers: namely, given two such bundles pi_1 and pi_2 as input, we can output a "bracket bundle" [pi_1,pi_2]. I will talk about this bracket bundle construction and a formula relating the Kontsevich's classes of [pi_1,pi_2] with those of pi_1 and pi_2. The main input of the proof is a new but very natural configuration space generalizing the Fulton-MacPherson configuration spaces. This is a work in progress joint with Robin Koytcheff and Sander Kupers.
11.00-12.00 Yi Xie (Peking University, CN)
Title: Stable parabolic bundles over complex curves and singular instanton Floer homology
Abstract: Stable parabolic bundles are objects in algebraic geometry which have been studied by many people. Singular Instanton Floer homology is an invariant of links in 3-manifolds introduced by Kronheimer and Mrowka, which has been used to solve many problems in the low dimensional topology. It turns out the two things are closely related: knowledge on the moduli space of stable parabolic bundles can help the calculation of singular instanton Floer homology. In this talk, we will give a precise description of the cohomology ring of the moduli space of rank 2 stable parabolic bundles over complex curves. Then we will derive all the “universal relations” for singular Instanton Floer homology. This is joint work with Boyu Zhang.
JUNE 06 (THURSDAY)
09.30-10.30 Honghao Gao (Tsinghua University, CN)
Title: Legendrian knots and Lagrangian fillings
Abstract: Legendrian knots and their exact Lagrangian fillings are central objects to study in low dimensional contact and symplectic topology. Therefore, it is an important question to classify exact Lagrangian fillings up to Hamiltonian isotopy. It is conjectured that this classification is controlled by a quiver and some derived algebraic structures. In this talk, I will review the historical developments, and explain the algebraic machinery to distinguish fillings. Then, I will discuss the ideas to obtain a subjectivity result, which involving new ideas such as understanding polygons on surfaces, quiver with potentials, etc. This is based on a joint work with Roger Casals.
11.00-12.00 Jun Zhang (University of Science and Technology of China, CN)
Title: Strong Arnold chord conjecture via normalized capacities
Abstract: In this talk, we will show how to prove the strong Arnold chord conjecture for a large family of star-shaped domains in R^4, which says that for any Legendrian knot on their contact boundaries, there exists a Reeb chord on the Legendrian knot with distinct endpoints. The proof heavily relies on the recent advances in various equivalences of symplectic capacities on monotone toric domains. A higher-dimensional version for star-shaped domains in R^{2n} also holds under a technical condition. This talk is based on joint work with Jungsoo Kang.
14.00-15.00 Siqi He (The Chinese Academy of Sciences ,CN)
Title: Z2 harmonic spinors and 1-forms part 1: introduction and motivations
Abstract: Z2 harmonic spinors and forms generalize quadratic differentials on Riemann surfaces to higher dimensions, establishing deep connections with gauge theory, low-dimensional topology, and calibrated geometry. Taubes' work indicates that Z2 harmonic 1-forms and spinors are the natural boundaries for various gauge theory equations, including those for flat SL(2,C) connections and the generalized Seiberg-Witten equations with two spinors. In this talk, we will introduce this topic and discuss known results by Takahashi, Parker, Walpuski, Doan, Donaldson, Haydys, Mazzeo, and others.
15.30-16.30 Greg Parker (Stanford University, USA)
Title: Z2 harmonic spinors and 1-forms part 2: deformation theory and gluing
Abstract: The study of Z2 harmonic spinors presents many novel analytic challenges. The main challenges are due to the presence of singular elliptic operators that are coupled to the geometry of a singular set, giving them a similar character to free-boundary problems. Studying deformations of the singular set leads to a hidden elliptic pseudodifferential operator governing the Fredholm theory of these objects. This talk will introduce the deformation theory and discuss its application to proving gluing results about moduli spaces of Z2 harmonic spinors and Seiberg-Witten monopoles.
JUNE 07(FRIDAY)
09.30-10.30 Yannick Sire (Johns Hopkins University,USA)
Title: Eigenvalue estimates and a conjecture of Yau
Abstract: I will describe various upper and lower bounds on the spectrum of the Laplace-Beltrami on Riemannian manifolds. The upper bounds led to some important results in spectral geometry establishing a link between the so-called conformal spectrum and branched minimal immersions into Euclidean spheres. I will then move to describe a conjecture by Yau on the first eigenvalue on minimal submanifolds of the sphere, which is known only for some examples. I will then present some recent results where we improve quantitatively the best known lower bound (in the general case) of Choi and Wang of the mid 80’s. I will address some open problems and possible generalizations of our argument.
11.00-12.00 Mingyang Li （The University of California, Berkeley, USA）
Title: Classification results for Hermitian non-Kahler gravitational instantons.
Abstract: We will discuss some classification results for Hermitian non-Kähler gravitational instantons. There are three main results: (1) Non-existence of certain Hermitian non-Kähler ALE gravitational instantons. (2) Complete classification for Hermitian non-Kähler ALF/AF gravitational instantons. (3) Non-existence of Hermitian non-Kähler gravitational instantons under suitable curvature decay condition, when there is more collapsing at infinity (ALG, ALH, etc.). These are achieved by a thorough analysis of the collapsing geometry at infinity and compactifications.
Popular Science talks（June 5 Wednesday afternoon）
Talk 1:
Title：肥皂膜的探索和极小曲面之美
Speaker：李琼玲（南开大学 陈省身数学研究所）
Abstract：我们通过对肥皂膜的讨论和实验来了解大自然如何选择它的形状，其中可以观察到许多非常微妙的数学原理，从而体会极小曲面之美。
Talk 2
Title：MATHEMATICS FOR AI - AND AI FOR MATHEMATICS
Speaker：Wilderich Tuschmann（KIT）
Abstract：Recent years have seen an enormous evolution of the use of Artificial Intelligence (AI) techniques in data-driven sciences, and Mathematics and its techniques and concepts constitutes the theoretical core of these developments, especially when considering, e.g., Machine Learning and the fundamental role nowadays played by Deep Neural Networks.
On the other hand, guiding and enhancing mathematicians' intuition, for centuries new mathematical theories have been constructed based on the study of examples and trying to find emerging patterns - just think of Carl Friedrich Gauss' tedious work of compiling lists of prime numbers which eventually made him come up with his famous theorem on their distribution, or, after computers were invented, their use in generating data to better understand, for example, the hitherto still unsolved 'Millenium Price Problem' Birch and Swinnerton-Dyer Conjecture, and here new approaches via AI can nowadays also be extremely helpful for further advancing mathematical progress.
In my talk, based on concrete real-world as well as 'pure math' examples and applications, I will shed further light on both sides of this precious medal.
Conference Venue
5^{th} Floor, SIENA NARADA GRAND HOTEL(青田西娜君澜大饭店)
Specific Conference Hall:
June 3^{rd} Morning Session: WENLAN HALL(文澜厅)
The Rest Sessions: WENTAO HALL(文涛厅)
Conference Designated Hotel
SIENA NARADA GRAND HOTEL(青田西娜君澜大饭店)
Address: Building 4,Baiyuecheng,Ou’nan Street, Qingtian County, Lishui City, Zhejiang Province(浙江省青田县瓯南街道百悦城4幢)
Dining Hall：
Breakfast: ALL DAY DINING(澜悦全日制餐厅)，4^{th} Floor, SIENA NARADA GRAND HOTEL
Lunch: ALL DAY DINING(澜悦全日制餐厅)，4^{th} Floor, SIENA NARADA GRAND HOTEL
Dinner: WENZE HALL(文泽厅)，5^{th} Floor, SIENA NARADA GRAND HOTEL
Banquet（For invited guests）:
18:00 June 3^{rd}
WENLAN HALL(文澜厅)，5th Floor, SIENA NARADA GRAND HOTEL