# 专题报告（7月6-9日）

**程哲驰(武汉大学)：A Symplectic Approach of Khovanov Homology
**

摘要：There are many different approaches of Khovanov homology. As for this talk, we will mainly be interested in a version from symplectic geometry, called symplectic Khovanov homology. It is conjecturally isomorphic to Khovanov homology. In 2020, Abouzaid and Smith proved this conjecture over characteristic-zero fields, while the more general cases remain open. In this talk, we start with comparing the gradings on symplectic Khovanov homology and Khovanov homology, and then discuss some recent progress on the conjecture.

**何东泰(同济大学)：Heegaard Floer homology and monodromy of open books
**

摘要：In this talk, we will discuss the relationship between Heegaard (Knot) Floer homology, contact structures, and the monodromy of open books. We will begin with an introduction to basic notions in contact topology, open book decomposition, with an emphasis on their relationship with Heegaard Floer homology. We will also discuss recent progress in this field, as well as application to knot concordance.

**何思奇(中科院数学所)：Z2 harmonic 1-forms: connections in topology and geometry**

摘要：Z2 harmonic spinors and forms extend the concept of quadratic differentials on Riemann surfaces to higher dimensions, creating significant links with gauge theory, low-dimensional topology, and calibrated geometry. According to Taubes, Z2 harmonic 1-forms serve as essential boundaries in various gauge theory equations, particularly in the context of flat SL(2,C) connections. In the first session, we will provide an overview of this field, highlighting contributions from Takahashi, Parker, Walpuski, Doan, Donaldson, Haydys, Mazzeo, Chen, and others. The second session will address a challenge question posed by Taubes-Wu concerning the existence and rigidity of the tangent cone model for Z2 harmonic 1-forms. We will discuss the application of finite group representation theory to this problem.

**谷世杰(东北大学)：Compactifications of manifolds**

摘要：In 1966, Larry Siebenmann once mused that his work (PhD thesis) was initiated at a time "when 'respectable' geometric topology was necessarily compact." That attitude has long since faded; today's topological landscape is filled with research in which noncompact spaces are primary objects. However, major successes in compactifying manifolds included here are fundamental to manifold topology: Stallings' characterization of Euclidean spaces, Siebenmann's collaring theorem, and our recent Gu-Guilbault's manifold completion theorem. In the first part, I will provide quick access to some of these results by weaving them together with common interpretations. In the second part, I will introduce several open questions on this topic. I will focus on clarifying the relationship between pseudo-collarability and Z-compactifiability, two main extensions on completable manifolds. I will construct counterexamples to the statement that Z-compactifiability implies pseudo-collarability. The constructions are based on knot theory and 4D topology. If time permits, I’ll show the reverse statement holds for manifolds of dimension at least six, i.e., pseudo-collarability implies Z-compactifiability.

**李友林(上海交通大学)：On contact solid tori in contact 3-manifolds**

摘要：Contact solid tori in contact 3-manifolds are closely related to Legendrian knots and Legendrian cable knots. In this talk, I will present several recent results concerning contact solid tori in contact 3-manifolds. This is joint work in progress with John Etnyre and Bulent Tosun.
预备报告题目：Convex surfaces in contact 3-manifolds

预备报告摘要：Convex surface theory is an essential tool in studying contact 3-manifolds. In this talk, I will briefly introduce the convex surface theory and demonstrate how it is used to study tight contact structures.

**潘宇(天津大学)：Legendrian knots and exact Lagrangian fillings**

摘要：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. On one hand, we use augmentations to build obstructions for exact Lagrangian cobordisms. On the other hand, we realize augmentations, which is an algebraic object, fully geometrically via exact Lagrangian surfaces.

**覃帆(北京师范大学)：Bracelets are theta functions for surface cluster algebras**

摘要：The skein algebra of a marked surface admits the basis of bracelet elements constructed by Fock-Goncharov and Musiker-Schiffler-Williams. As a cluster algebra, it also admits the theta basis from the cluster scattering diagram by Gross-Hacking-Keel-Kontsevich. In a joint work with Travis Mandel, we show that the two bases coincide except for the once-punctured torus. Long-standing conjectures on strong positivity and atomicity follow as corollaries.预备报告题目：Visualizing cluster algebras through topological models

预备报告摘要：Cluster algebras are algebras with a rich combinatorial structure. They are ubiquitous in mathematics. In this talk, we will illustrate these algebras and associated higher structures using topological models on surfaces.

**杨璟玲(厦门大学)：Knot concordance, slice genus and Heegaard Floer homology**

摘要：In this talk, we will discuss the 4-dimensional properties of knots, knot concordance and knot slice genus, which play central roles in low-dimensional topology. Heegaard Floer ho- mology has proved to be an effective tool in studying low-dimensional topology, particularly in advancing the understanding of knot concordance. We will begin with an expository introduction to some elementary notions, followed by a review of results in knot concordance stemming from Heegaard Floer theory. We will also discuss our recent progress in this field. This is a joint work with Zhongtao Wu.

**张俊(中国科学技术大学)：Givental’s non-linear Maslov index via Floer cones**

摘要：In this talk, we will present how to apply a recently-developed Floer theory on a fillable contact manifold, called the contact Hamiltonian Floer homology, to generate a homological machinery that replaces the classical Givental non-linear Maslov indices. As a key step, we will emphasize the role of the homological mapping cone from this Floer theory (called a Floer cone) and its fundamental role in reflecting local data of periodic orbits. As an application, the multiplicity of translated points, serving as a natural generalization of fixed points in contact Hamiltonian dynamics, will be deduced. This talk is based on joint work with Dylan Cant and Igor Uljarevic. 预备报告题目：Contact geometry and its associated dynamics.

预备报告摘要：In this preliminary talk, we will recall some general background on contact geometry, mainly focusing on the Hamiltonian dynamics on a contact manifold. In particular, Reeb dynamics and various studies on closed Reeb orbits will be elaborated; orderability from Eliashberg-Polterovich, together with its most updated progress in terms of the metric geometry, will also be covered. If time permits, we will outline how the Hamiltonian dynamics behave in the relative situation, that is, on Legendrian submanifolds or more generally on contact coisotropic submanifolds.

# 学生报告

**代丕孟：
Simplicial complex with extremal total betti number**

摘要：We determine which simplicial complexes have the maximum or minimum sum of
Betti numbers and bigraded Betti numbers with a given number of vertices in each
dimension.

**尚鉴桥：
一个 Morse 同调的引理**

摘要：（引理）考虑流形 M 与子流形 N，以及 M 上的 Morse 函数 f。如果 f 在 M 和 N 上拥有
相同的奇异点与指标，那么 M 与 N 的 Morse 同调相同。
我们将证明该引理，用其解释 Lagrange 乘子法的 Morse 理论, Viterbo 同构等定理为何正确。之后我会从其出发，解释进一步对辛同调能猜想什么。

**刁文杰：Square knots, surgery on links and homotopy 4-spheres**

摘要：While the theory of Dehn surgery on knots has been thoroughly developed over
the past forty years, much less seems to be known about Dehn surgery on links. This
expository talk will review recent results of Gompf-Scharlemann-Thompson and
Meier-Zupan on n-component links in S^3 with a Dehn surgery realizing #n(S^1×S^2)
and its relation to the Smooth 4-Dimensional Poincare conjecture and the Generalized
Property R conjecture.

**王云
杰：Short closed geodesics with self-intersections on hyperbolic surfaces**

摘要：In this talk，I will briefly introduce the counting number problem
on hyperbolic surfaces. We will describes the exact asymptotic behavior of
the minimal ratio between length and rooting of self-intersection number
geodesics on compact hyperbolic surfaces, as a function on the moduli
space in terms of their systole length.
This is a joint work with Lizhi Chen.

**Wenbo
Liao：An Alexander Polynomial for Spatial Graphs and the trapezoidal conjecture**

摘要：We introduce an Alexander polynomial for spatial graphs by generalizing
Kaufmann states sum. Let G ⊂S^3 be a spatial graph and G be any planar projection
of G. We define the Kaufmann states of G and show the state sum is independent of
the choice of G so that we get a well-defined invariant of G, called Alexander
polynomial. By using this new invariant, we give a necessary condition for spatial
graphs being planar graphs and give an intrinsic invariant of graphs. Also, we
briefly introduce our recent proof on the trapezoidal conjecture for planar graphs
and relate it to the trapezoidal conjecture for alternating knots.

**张昊航：Rigidity of the grid graph**

摘要：Inspired by asymptotically flat manifolds, we introduce the concept of
asymptotically flat graphs. We formulate the discrete positive mass conjecture based
on the scalar curvature in the sense of Ollivier curvature, and prove the positive
mass theorem for asymptotically flat graphs that are combinatorially isomorphic to
grid graphs. We prove a weaker version of the positive mass conjecture: an
asymptotically flat graph with non-negative Ricci curvature is isomorphic to the
standard grid graph. Hence the topology structure of an asymptotically flat graph
is determined by the curvature condition, which is a discrete analog of the rigidity
part for the positive mass theorem. The key tool for the proof is the discrete
harmonic function of linear growth associated with the salami structure.链接:
https://arxiv.org/abs/2307.08334