召集人：阮勇斌院士（浙江大学，教授）、刘毅（北京大学，教授）、徐宙利（加州大学洛杉矶分校，教授）

时间：2025.09.14—2025.09.20

会议日程安排

2025 年 9 月 15日，星期一

2025 年 9 月 16  日，星期二

2025 年 9 月 17 日，星期三

2025 年 9 月 18 日，星期四

2025 年 9 月 19 日，星期五

报告题目和摘要  

2025 年9月15日（星期一）

Hempel pairs and Turaev Viro invariants

刘毅（北京大学）

Abstract: Hempel pairs are periodic surface bundles with profinitely isomorphic fundamental groups. In this talk, I will discuss whether Turaev--Viro invariants distinguish such pairs. I will explain some motivation of this work and discuss further questions.

Legendrian knots, Possion variety and quantization

高鸿灏（清华大学）

Abstract: Legendrian knots lie in the overlap of low dimensional topology and contact topology. One of the most robust invariant for Legendrian knots is the Chekanov dg algebra, an invariant defined via Floer theory. In this talk, we will discuss a possion structure on the variety of points of the non-commutative dg algebra. The definition comes from symplectic field theory. Further, the possion structure leads to a quantization of the algebra. This is a joint work with Casals, Ng, Shen, Weng and Zalsow.

C^0-closedness of Symp_0(X)

吴惟为（浙江大学）

Abstract: The C^0 topology of the symplectomorphism groups has lots of mysterious basic questions.  The famous symplectic rigidity theorem says that, given any symplectic manifold X, the symplectomorphism group Symp(X) is closed in Diff(X) with respect the the C^0 topology.  The question of whether Ham(X) is C^0-closed in Symp_0(X) is the content of the famous C^0 flux conjecture.  However, the relation between Symp(X) and Symp_0(X) is largely unexplored.  Using Floer theory, Jannaud proved that iterations of a Dehn twist cannot lie in the closure of Symp_0(X) for some Liouville domains.

In this talk, we will present a proof of the closedness of Symp_0(X) in Symp(X) when X is a log Calabi-Yau surface of type D, in the sense of Li-Li-Wu.  For these symplectic manifolds, it was previously known that Symp_h(X) is a subset of Diff_0(X).  The key to our approach is to apply the techniques of J-holomorphic foliations and inflations to obtain necessary C^0-estimates.  This is a joint work with Marcelo Atallah and Cheuk-Yu Mak. 

2025 年 9月16日（星期二）

几何三维流形公度类的手性

王诗宬（北京大学）

Abstract: 称流形有(无)手性，若 M 上无(有)反定向自同胚。手性是三维空间的一个基本现象。两个流形叫可公度的，若他们有公共的有限覆叠。称一个流形公度类无手性，若其中含有无手性流形。

我们将报告田野、尹航、王中子最近关于几何三维流形公度类手性的研究。该研究表明，一方面流形的手性常常根植于它的公度类或几何中，另一方面公度类手性的研究常常与代数与数论有更为深刻的联系。

Poincaré polynomials of moduli spaces of 1-dimensional sheaves on the projective plane

邬龙挺（南方科技大学）

Abstract: The geometry of moduli spaces of one-dimensional sheaves on the projective plane has attracted a lot of study recently. In this talk, I will give a new calculation of the Betti numbers of the moduli spaces of one-dimensional sheaves on the projective plane using Gromov-Witten invariants of local P^2 and local curves. The new calculation is based on the refined sheaves/GW correspondence established by Bousseau and all genus local/relative correspondence given by Bousseau-Fan-Guo-Wu. It can be used to prove the divisibility property of Poincaré polynomials of moduli spaces of one-dimensional sheaves on projective plane conjectured by Choi-van Garrel-Katz-Takahashi, and can also be used to determine the leading Betti numbers. Some conjectures concerning the higher range Betti numbers will be proposed if time permits. This is based on a joint work with Shuai Guo and Miguel Moreira.

Morse theory, Floer homology, and Hecke algebra

袁天宇（东方理工大学）

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, Roman Krutowski, and Yin Tian.

2025年9月17日（星期三）

Title: Higgs bundles over non-compact surfaces

李琼玲（南开大学）

Abstract: The non-abelian Hodge correspondence establishes a homeomorphism between the character variety of a surface group and the moduli space of polystable Higgs bundles over a compact Riemann surface. A central step in this correspondence is solving the Hitchin equation to find a harmonic metric. In this talk, I will present new results on the existence and uniqueness of harmonic metrics for Higgs bundles over non-compact surfaces, extending the classical theory beyond the compact setting.

Embedding groups into acyclic groups

伍晓磊（复旦大学）

Abstract: We first discuss various embedding results for groups in the literature. Then we talk about how could one embed a group of type F_n into an acyclic group of type F_n. The embedding we have uses the labelled Thompson group which goes back to Thompson's Splinter group in the 1980s. We explain how one can show that the labeled Thompson groups are always acyclic. This also allows us to build acyclic groups of type F_n but not F_{n+1} for any n. If time permited, I will also discuss related results in the simple setting using the twisted Brin--Thompson groups. This is based on a joint work with Martin Palmer.

Some results on cobordism between $3$-manifolds with constraints on $H_1$

王中子（北京大学）

Abstract: We call two closed oriented connected 3-manifolds $Y_1$ and $Y_2$ are $H_1$-injective cobordant if there exists a compact oriented $4$-manifold $W$ such that $\partial W=Y_1\cup -Y_2$ ( where $-Y$ is the orientation reversal of $Y$) and the inclusion maps $Y_i\to W$ induces injections on $H_1$.

If $Y$ is the boundary of a compact oriented $4$-manifold $W$ such that the inclusion map $Y\to W$ is injective on $H_1$, then we say $Y$ is $H_1$-injective null-cobordant.

We proved that the lens spaces $L(p,1)$ where $p$ is a prime number are not $H_1$-injective null-cobordant. We also proved that all $3$-manifolds with $H_1=0, \mathbb{Z}$ or $\mathbb{Z}^2$ are $H_1$-injective null-cobordant.

This is a joint work with Jianfeng Lin.

2025年9月18日（星期四）

Periodicities of higher real K-theories

XiaoLin Danny Shi（美国华盛顿大学）

Abstract: Historically, topological K-theory and its Bott periodicity have been very useful in solving key problems in algebraic and geometric topology. In this talk, we will explore the periodicities of higher real K-theories and their roles in several contexts, including Hill--Hopkins--Ravenel’s solution of the Kervaire invariant one problem. We will prove periodicity theorems for higher real K-theories at the prime 2 and show how these results feed into equivariant computations. We will then use these periodicities to measure the complexity of the RO(G)-graded homotopy groups of Lubin--Tate theories and to compute their equivariant slice spectral sequences. This is joint work with Zhipeng Duan, Mike Hill, Guchuan Li, Yutao Liu, Guozhen Wang, and Zhouli Xu

Algebraic topology of 24 dimensional string manifolds

黄瑞芝（中国科学院）

Abstract: String structures can be viewed as a lift of spin structures, and play a fundamental role in both mathematics and theoretical physics. In particular, they are deeply connected to the Atiyah-Singer index theory. Beyond their physical relevance, there has been enduring interest in the geometry and topology of string manifolds. Among them, 24-dimensional string manifolds exhibit especially rich and intriguing structure.

In this talk, we will explore the algebraic-topological aspects of 24-dimensional string manifolds, drawing on work by Hirzebruch, Ochanine, Landweber-Stong, Wall, Mahowald-Hopkins, Chen-Han, as well as recent joint work of mine with Fei Han.

On the q-de Rham Operators

郭靖邦（复旦大学）

Abstract: The q-de Rham operators, which deform the ordinary algebraic de Rham operators, are important tools to understand cohomology theories in the p-adic settings.

In this talk, the theory of prismatic cohomology will be briefly reviewed through particular Hopf algebroids, whose dual algebras can be regarded as algebras of q-de Rham operators, providing a perspective to both classification and computation. In particular, for certain polynomial rings, explicit complexes formed with q-de Rham operators can be written down for computation.

Through the motivic style filtrations relating prismatic cohomology and topological cyclic homology, and the cyclotomic trace relating topological cyclic homology and algebraic K-theory, the q-de Rham operators become potential tools to understand the algebraic K-theory of certain rings.

2025年9月19日（星期五）

Exponential volumes of moduli spaces of hyperbolic surfaces

孙哲（中国科学技术大学）

Abstract: Mirzakhani found a remarkable recursive formula for the volumes of the moduli spaces of the hyperbolic surfaces with geodesic boundary, and the recursive formula plays very important role in several areas of mathematics: topological recursion, random hyperbolic surfaces etc.   We consider some more general moduli spaces M_S(K,L) where the hyperbolic surfaces would have crown ends and horocycle decorations at each ideal points. But the volume of the space M_S(K,L) is infinite when S has the crown ends. To fix this problem, we introduce the exponential volume form given by the volume form multiplied by the exponent of a canonical function on M_S(K,L).   

We show that the exponential volume is finite. And we prove the recursion formulas for the exponential volumes, generalising Mirzakhani's recursions for the volumes of moduli spaces of hyperbolic surfaces. We expect the exponential volumes are relevant to the open string theory. This is a joint work with Alexander Goncharov.

Composed Dehn twist exact sequence through A infinity n-modules

张硕（中国科学院）

Abstract: We prove the quilted Floer cochain complexes form A infinity n-modules over the Fukaya category of Lagrangian correspondences.  Then we prove that when we restrict the input to mapping cones of product Lagrangians and graphs, the resulting bar-type complex can be identified with bar complex from ordinary Floer theory.  As an application we use a family version of quilt unfolding argument to prove two long exact sequences conjectured by Seidel that relates the Lagrangian Floer cohomology of a collection of (possibly intersecting) Lagrangian spheres and the fixed point Floer cohomology of composition of Dehn twists along them.

The tame Deligne-Simpson problem

舒成（西湖大学）

Abstract: We solve the long-standing problem of Deligne-Simpson: given conjugacy classes $(C_j)_{1\le j\le k}$ of invertible matrices of rank $n$, do there exist $A_j\in C_j$ such that (1) $A_1\cdots A_k=\Id$ and (2) there is no nontrivial proper subspace of $\mathbb{C}^n$ that is preserved by every $A_j$? A conjectural necessary and sufficient condition on $(C_j)_j$ in terms of certain Kac-Moody root systems was proposed by Crawley-Boevey, and the sufficiency statement was later proved in his joint work with Shaw. Our main result proves the necessity statement and the method is a combination of nonabelian Hodge theory and variation of stability conditions.