**Past Talks:**

**Large-scale geometry of the saddle connection graph - Robert TANG, Xi'an Jiaotong-Liverpool University (2022-05-24)**

For a translation surface, the associated saddle connection graph has saddle connections as vertices, and edges connecting pairs of non-crossing saddle connections. This can be viewed as an induced subgraph of the arc graph of the surface. In this talk, I will discuss both the fine and coarse geometry of the saddle connection graph. We show that the isometry type is rigid: any isomorphism between two such graphs is induced by an affine diffeomorphism between the underlying translation surfaces. However, the situation is completely different when one considers the quasi-isometry type: all saddle connection graphs form a single quasi-isometry class. We will also discuss the Gromov boundary in terms of foliations. This is based on joint work with Valentina Disarlo, Huiping Pan, and Anja Randecker.

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**Notes**

**Symmetries of exotic aspherical manifolds - Mauricio BUSTAMANTE, Universidad Católica de Chile (2022-05-19)**

Let W be a closed smooth n-manifold and W' a manifold which is homeomorphic but not diffeomorphic to W. In this talk I will discuss the extent to which W' supports the same symmetries as W when W is a n-torus or a hyperbolic manifold, and W' is the connected sum of W with an exotic n-sphere. As a sample of results, I will indicate how to classify all finite cyclic groups that act freely and smoothly on an exotic n-torus. For hyperbolic manifolds W, I will show how to produce examples of W' which admit no nontrivial smooth action of a finite group, while Isom(W) is arbitarily large. This is joint work with Bena Tshishiku.

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**A distance on Teichmueller space via renormalized volume - Hidetoshi MASAI 正井 秀俊, Tokyo Tech (2022-05-10)**

In this talk, we consider volumes of hyperbolic 3-manifolds and construct a new distance on the Teichmueller space of a closed surface of genus >1. We will compare the new distance with other known distances: Teichmueller distance, Weil-Petersson distance. If time permits, I would also like to talk about several questions about the new distance. This talk is based on the preprint https://arxiv.org/abs/2108.06059.

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**Connectivity of the space of pointed hyperbolic surfaces - Sangsan (Tee) WARAKKAGUN, BIMSA (2022-04-26)**

We consider the space of all complete hyperbolic surfaces with basepoint equipped with the pointed Gromov-Hausdorff topology. In this talk, I will begin by motivating this topology and reviewing basic surface hyperbolic geometry. Then, I will describe certain deformations on a hyperbolic surface and concrete geometric constructions which are used to show that the space is globally path-connected and is locally weakly connected at points whose underlying surfaces are either the hyperbolic plane or hyperbolic surfaces of the first kind.

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**Notes**

**Wilson lines and the A=U problem for the moduli spaces of G-local systems, Tsukasa ISHIBASHI 石橋 典, Tohoku U (2022-04-21)**

The moduli space of decorated twisted G-local systems on a marked surface, originally introduced by Fock--Goncharov, is known to have a natural cluster K_2 structure. In particular, we have a canonically defined cluster algebra A and an upper cluster algebra U inside its field of rational functions. In order to investigate the structure of the function ring of that moduli space, we introduce the Wilson lines valued in the simply-connected group G, which are “framed versions” of those studied by myself and Hironori Oya. We see that the function ring of the moduli space is generated by the matrix coefficients of Wilson iines, and some of them are cluster monomials. As an application, we prove that both A and U coincide with the function ring. Time permitting, I will also mention some relations to the skein theory. This talk is based on a joint work with Hironori Oya and Linhui Shen.

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**Notes**

**Homological stability for the ribbon Higman-Thompson groups - Xiaolei WU 伍晓磊, SCMS Fudan (2022-04-12)**

Abstract: I will start the talk with basics about Higman-Thompson groups and then introduce its braided version and ribbon version.I will build a geometric model for the ribbon Higman-Thompson groups, namely as a nice subgroup for the mapping class group of a disk minus a Cantor set. We use this model to prove that the ribbon Higman-Thompson groups satisfy homological stability. This can be treated as an extension of Szymik-Wahl's work on homological stability for the Higman-Thompson groups to the surface setting. This is a joint work with Rachel Skipper.

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**Deformation space of circle patterns** - Waiyeung LAM 林偉揚, BIMSA (2022-03-29)

William Thurston proposed regarding the map induced from two circle packings with the same tangency pattern as a discrete holomorphic function. A discrete analogue of the Riemann mapping is deduced from Koebe-Andreev-Thurston theorem. One question is how to extend this theory to Riemann surfaces and relate classical conformal structures to discrete conformal structures. Since circles are preserved under complex projective transformations, we consider circle packings on surfaces with complex projective structures. Kojima, Mizushima and Tan conjectured that for a given combinatorics the deformation space of circle packings is diffeomorphic to the Teichmueller space. In this talk, we explain how discrete Laplacian is used to prove the conjecture for the torus case and its connection to Weil-Petersson geometry.

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**Notes**

**Filtration of cohomology via symmetric semisimplicial spaces - Oishee BANERJEE, HCM Uni. Bonn (2022-03-22)**

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**The mapping class group of manifolds which are like projective planes - Yang SU 苏阳, CAS (2022-03-15)**

A manifold which is like a projective plane is a simply-connected closed smooth manifold whose homology equals three copies of Z. In this talk I will discuss our computation of the mapping class group of these manifolds, as well as some applications in geometry. This is a joint work with WANG Wei from Shanghai Ocean University.

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**Nonexistence of symplectic structures on certain family of 4-manifolds - Jianfeng LIN 林剑锋, Tsinghua (2022-03-08)**

Let Symp(X) be the group of symplectomorphisms on a symplectic 4-manifold X. It is a classical problem in symplectic topology to study the homotopy type of Symp(X) and to compare it with the group of all diffeomorphisms on X. This problem is closely related to the existence of symplectic structures on smooth families of 4-manifolds. In this talk, we will discuss the proof of following results: (1) For any X that contains a smoothly embedded 2-sphere with self-intersection -1 or -2, there exists a loop of self-diffeomorphisms on X that is not homotopic to a loop of symplectomorphisms. (2) Consider a family of 4-manifolds obtained by resolving an ADE singularity using a hyperkahler family of complex structures, this family never support a family symplectic structure in a constant cohomology class. (3) For any non-minimal symplectic 4-manifold whose positive second-betti number does not equal to 3, the space of symplectic form is not simply connected. The key ingredient in the proofs is a new gluing formula for the family Seiberg-Witten invariant.

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**Notes**

**Khovanov skein homology for links in the thickened torus - Yi XIE 谢羿, PKU, BICMR (2022-03-01)**

Asaeda, Przytycki and Sikora defined a generalization of Khovanov homology for links in thickened compact surfaces. In this talk I will show that the Asaeda-Przytycki-Sikora homology detects the unlink and torus links in the thickened torus. This is joint work with Boyu Zhang.

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This seminar series is a continuation of the GeoTop Seminars from the 2022 Autumn/Fall semester, and here are the talks from then:

**Hyperbolic Geometry and Quantum Invariants - Tian YANG 杨田, Texas A&M (2021-12-20)**

There are two very different approaches to 3-dimensional topology, the hyperbolic geometry following the work of Thurston and the quantum invariants following the work of Jones and Witten. These two approaches are related by a sequence of problems called the Volume Conjectures. In this talk, I will explain these conjectures and present some recent joint works with Ka Ho Wong related to or benefited from this relationship.

**Additional info**: 2021-12-20 YANG notes

**Norms on cohomology of non-compact hyperbolic 3-manifolds, harmonic forms and geometric convergence - Hans Xiaolong HAN 韩肖垄, Tsinghua (2021-12-13, part 2)**

(This is part 2, and continues from where the previous talk left off)

We will talk about generalizations of an inequality of Brock-Dunfield to the non-compact case, with tools from Hodge theory for non-compact hyperbolic manifolds and recent developments in the theory of minimal surfaces. We also prove that their inequality is not sharp, using holomorphic quadratic differentials and recent ideas of Wolf and Wu on minimal geometric foliations. If time permits, we will talk about some results concerning the growth of L2 norm/Thurston norm for a sequence of closed hyperbolic 3-manifolds converging geometrically to a cusped manifold, using Dehn filling and minimal surface.

**Norms on cohomology of non-compact hyperbolic 3-manifolds, harmonic forms and geometric convergence - Hans Xiaolong HAN 韩肖垄, Tsinghua (2021-12-06, part 1)**

We will talk about generalizations of an inequality of Brock-Dunfield to the non-compact case, with tools from Hodge theory for non-compact hyperbolic manifolds and recent developments in the theory of minimal surfaces. We also prove that their inequality is not sharp, using holomorphic quadratic differentials and recent ideas of Wolf and Wu on minimal geometric foliations. If time permits, we will talk about some results concerning the growth of L2 norm/Thurston norm for a sequence of closed hyperbolic 3-manifolds converging geometrically to a cusped manifold, using Dehn filling and minimal surface.

**Augmentations from Legendrian knots - Tao SU 苏桃, Tsinghua (2021-11-29)**

In this talk, I will tell a story about Legendrian knots, with a focus on the associated Chekanov-Eliashberg DGAs and their augmentations. First, I will introduce the Chekanov-Eliashberg DGA. It’s a Legendrian isotopy invariant up to homotopy equivalence, which admits two equivalent descriptions: counting of pseudo-holomorphic disks, and combinatorics. Second, I will discuss the gluing property of the Chekanov-Eliashberg DGA, induced by cutting the Legendrian knot front diagram into elementary pieces. Finally, I will give an application of this gluing property: counting augmentations gives a state-sum Legendrian isotopy invariant, i.e. the ruling polynomial. Time permitting, I will also mention a second application in my recent work, concerning part of the geometric P=W conjecture.

**How tight can a contact manifold be? - Zhengyi ZHOU 周正一, AMSS Chinese Academy of Science (2021-11-22)**

The fundamental dichotomy of overtwisted v.s. tight in contact topology asserts that contact topology of overtwisted structures can be completely “understood” in a topological manner. On the other hand, the tight contact structures form a richer and more mysterious class. In this talk, I will explain how to use rational symplectic field theory to give a hierarchy on contact manifolds to measure their “tightness”. This is a joint work with Agustin Moreno.

**Proofs of Mostow Rigidity Theorem - Qing LAN 蓝青, Tsinghua (2021-11-22)**

In this talk I will sketch 2 proofs of Mostow rigidity, which essentially states that the geometry of a closed hyperbolic manifold of dimension greater than two is determined by the fundamental group. I will talk about a proof using ergodic theory and another proof using Gromov norm.

**Extended graph manifolds, and Einstein metrics - Luca DI CERBO, University of Florida (2021-11-04)**

In this talk, I will present some new topological obstructions for solving the Einstein equations (in Riemannian signature) on a large class of closed four-manifolds. I will conclude with some tantalizing open problems both in dimension four and in higher dimensions.

**Trisection invariants of 4-manifolds from Hopf algebras - Xingshan CUI 崔星山, Purdue (2021-10-25)**

The Kuperberg invariant is a topological invariant of closed 3-manifolds based on finite-dimensional Hopf algebras. Here we initiate the program of constructing 4-manifold invariants in the spirit of Kuperberg's 3-manifold invariant. We utilize a structure called a Hopf triplet, which consists of three Hopf algebras and a bilinear form on each pair subject to certain compatibility conditions. In our construction, we present 4-manifolds by their trisection diagrams, a four-dimensional analog of Heegaard diagrams. The main result is that every Hopf triplet yields a diffeomorphism invariant of closed 4-manifolds. In special cases, our invariant reduces to Crane-Yetter invariants and generalized dichromatic invariants, and conjecturally Kashaev's invariant. As a starting point, we assume that the Hopf algebras involved in the Hopf triplets are semisimple. Time permitting, we also sketch an ongoing effort to generalize the invariant using non-semisimple Hopf algebras. The generalized invariant is defined on 4-manifolds with a choice of spin^c structure. We expect the generalized invariant is more sensitive to extract information about 4-manifolds.

**Let's stretch some hyperbolic surfaces! - Yi HUANG 黄意, Tsinghua (2021-10-18)**

In an unpublished preprint, Thurston looked into the Lipschitz theory of hyperbolic surfaces and built from scratch a beautiful theory tying together stretch maps and the lengths of simple closed geodesics on hyperbolic surfaces. We hope to give a gentle introduction to this theory, and to introduce some modern explorations along this theme.

**Turning smooth 4-manifolds into maps between spheres - Jianfeng LIN 林剑锋 (2021-10-04)**

In the past 40 years, studying smooth structures on 4-manifolds has been an important topic in low dimensional topology. In this talk, I will talk about the celebrated Bauer-Furuta invariant of 4-manifolds. In particular, I will dicuss a technique called the ``finite dimensional approximation'' which is a general procedure that turns a nonlinear elliptic PDE into a map between two (finite dimensional) spheres. This allows us to use powerful tools from equivariant stable homotopy theory to attack hard problems in 4-dimensional topology. I will also talk about some recent applications of this invariant on exotic diffeomorphisms and exotic embeded surfaces in 4-manifolds.

**Topological complexity of enumerative problem - Weiyan CHEN 陈伟彦 (2021-09-27)**

Topological complexity measures how difficult is it to find solutions of a problem using an algorithm. It has been extensively studied in the context of topological robotics and superposition of algebraic functions. In this talk, I will propose a new research direction of determining topological complexity of enumerative problems in algebraic geometry. As an example, I will talk about our recent theorem on the topological complexity of finding flex points on smooth cubic plane curves. This talk is based on joint work with Zheyan Wan 万喆彦. I will try to make the talk accessible to undergraduate students who have taken a course in algebraic topology.