**Some connections between topology and number theory**

**Shicheng WANG 王诗宬, PKU (2023-12-12)**

We will discuss some connections between topology and number theory inspired by the studies of mapping degrees and achirality of manifolds.

Time: 2023-12-12, TUESDAY 16:00-17:00

Place: Shuangqing Complex 双清综合楼, C654

Zoom meeting ID: 405 416 0815, pw: 111111

**Past Talks:**

**Generic 3-manifolds are hyperbolic - ****Wenyuan YANG 杨文元, PKU (2023-12-05)**

In this talk, we first introduce various models to study what a generic 3-manifold looks like. We then focus on the Heegaard splitting model of 3-manifolds, equipped with geometric complexity using Teichmuller metric. The main result is that the Hempel distance of a generic Heegaard splitting goes linearly to the infinity. In particular, generic 3-manifolds are hyperbolic in this model. This represents the joint work with Suzhen Han and Yanqing Zou.

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**On pseudo-collarability and Z-compactifiablity of manifolds - ****Shijie GU 谷世杰, Northeastern University (2023-11-28)**

Siebenmann's landmark 1965 dissertation established conditions for compactifying open high-dimensional manifolds by adding boundaries, a process termed 'completion' or 'collaring'. Nearly six decades later, Gu-Guilbault broadened this scope to include noncompact manifolds with boundaries, offering a complete characterization of completable manifolds. However, the emergence of exotic universal covering spaces and shifts towards topics like the Borel and Novikov conjectures and geometric group theory, where fundamental groups at infinity are unstable, has necessitated extensions to the completion of manifolds. Key among these are pseudo-collarability, introduced by Guilbault in 2000, and Z-compactifiability, dating back to Anderson's 1967 work on infinite-dimensional manifolds. This talk aims to address the implication between these two concepts. Specifically, we will show that a well-established set of conditions proposed by Chapman and Siebenmann in 1976 assures Z-compactifiability of manifolds. Time permitting, we will also discuss some applications to the Borel conjecture.

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**Points on spheres, character varieties, and Stokes matrices - ****Yu-Wei FAN 范祐维, YMSC (2023-11-21)**

The moduli spaces of points on n-spheres carry natural actions of braid groups. For n=0, 1, and 3, we prove that these symmetries extend to actions of mapping class groups of certain positive genus surfaces, through exceptional isomorphisms with certain moduli of local systems. Moreover, the isomorphisms map the Coxeter invariants of points on spheres to the boundary monodromy of the corresponding local systems. We also apply this connection to the study of Stokes matrices and exceptional collections. This talk is based on a prior joint work with Junho Peter Whang.

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**Horospheres, Lipschitz maps, and laminations - ****James FARRE, MPI Leipzig (2023-11-14)**

Every horocycle in a closed hyperbolic surface is dense, and this has been known since the 1940's. We study the behavior of horocycle orbit closures in Z-covers of closed surfaces, and obtain a fairly complete classification of their topology and geometry. The main tool is a solution of a surprisingly delicate geometric optimization problem: finding an optimal Lipschitz map to the circle and the associated lamination of maximal stretch. I will focus on a novel construction of these optimal Lipschitz maps using the orthogeodesic foliation construction. This is joint work with Yair Minsky and Or Landesberg featuring joint work with Aaron Calderon.

**Wrapped Floer theory for Lagrangian fillings**** - ****Yu PAN 潘宇, Tianjin University 天津大学 (2023-11-07)**

Lagrangian fillings are key objects in symplectic geometry. Wrapped Floer theory can be used to show some rigidity property of embedded Lagrangian fillings. We extend the wrapped Floer theory to immersed Lagrangian fillings and obtain lower bounds of double points of immersed Lagrangian disk fillings.

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**Topological classification of Bazaikin spaces - ****Wen SHEN 沈文, Capital Normal University 首都师范大学 (2023-10-31)**

Manifolds with positive sectional curvature have been a central object dates back to the beginning of Riemannian geometry. Up to homeomorphism, there are only finitely many examples of simply connected positively curved manifolds in all dimensions except in dimension 7 and 13, namely, Aloff-Wallach spaces and Eschenburg spaces in dimension 7, and the Bazaikin spaces in dimension 13. The topological classification modelled on the 7-dimensional examples has been carried out by Kreck-Stolz which leads to a complete solution for the Aloff-Wallach spaces. The main goal of this report is to discuss the topological classification problem of the Bazaikin spaces.

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**Systolic Inequality and Topological Complexity of Manifolds - ****Lizhi CHEN 陈立志, Lanzhou University 兰州大学 (2023-10-24)**

The systole of a closed Riemannian manifold is defined to be the length of a shortest noncontractible loop. Gromov's systolic inequality relates systole to volume, which is a curvature free inequality. Gromov proved that systolic inequality holds on closed essential manifolds. Gromov's further work indicates that systolic inequality is related to topological complicatedness of manifolds. Analogously, Berger's embolic inequality is another curvature free inequality, also reflecting topological properties. In this talk, we introduce some new developments concerning the relation between these two curvature free inequalities and the topology of manifolds.

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**Notes**

**Characteristic numbers, Jiang subgroup and non-positive curvature - ****Ping LI 李平, Fudan (2023-10-16)**

A sufficient condition in terms of Jiang subgroup is presented for the vanishing of signature and arithmetic genus. Along this line, some progress can be made on a question of Farrell and a complex version's Hopf conjecture.

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**Masur's criterion does not hold in the Thurston metric - ****Ivan TELPUKHOVSKIY, YMSC (2023-10-10)**

Masur proved that if a Teichmüller geodesic is recurrent in the moduli space then it has a uniquely ergodic vertical foliation. This is a basic result in Teichmüller dynamics. However, it was shown that an analogue of Masur's criterion is not true in the Weil-Petersson metric (Brock-Modami) and in the Thurston metric (T.). In the talk, I will explain why it fails in the Thurston metric. Along the way we will discuss how to combinatorially construct minimal geodesic laminations on the surface

that are not uniquely ergodic.

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**Notes**

**The Links–Gould Invariants of links as classical generalizations of the Alexander polynomial - ****Ben-Michael KOHLI (2023-09-26)**

The Links-Gould invariants of links are a family of two variable polynomial quantum link invariants built using Hopf superalgebras U_q(gl(m|n)).

However, we now know that the Alexander invariant of links can be recovered by evaluating the Links-Gould invariants in several different ways, using works by De Wit-Ishii-Links, and more recently Kohli and Patureau-Mirand. Therefore, all the information that the Alexander polynomial carries is also contained in the Links-Gould invariants of links. That includes the homological information that the Alexander invariant gives about the link: genus bound, fiberdness, …

Hence, one can wonder whether the Links-Gould invariants of links could generalize some of the classical properties of the

Alexander polynomial. It seems that it should be the case, and recent work with Geer, Patureau-Mirand and Tahar hints further to a classical construction for these quantum invariants.

**Watch**

**String topology, cyclic homology, and the Fukaya A-infinity algebra - ****Yi WANG, Stony Brook (2023-06-28)**

This talk is devoted to giving a formulation of Lagrangian Floer theory in terms of chain level string topology, which realizes a proposal of K. Fukaya. First, I will describe a new chain model of the (based and free) loop space of a path-connected topological space X, which can be viewed as a generalization of classical theorems of J. F. Adams and K-T Chen. Then I will combine this model with a Jones' type theorem on cyclic homology and S^1 equivariant homology, as well as K. Irie's work on string topology, to describe chain level string topology operations in the S^1-equivariant setting, in particular, chain level string bracket (cyclic loop bracket). Finally, I will use this chain model to lift the Fukaya A-infinity algebra of a Lagrangian submanifold L to a Maurer-Cartan element in the dg Lie algebra of cyclic invariant chains on the free loop space of L, and discuss possible applications in symplectic topology.

**Notes**

**Edge-to-Edge Tilings of the Sphere by Congruent Polygons - Min YAN 严民, Hong Kong University of Science and Technology (2023-06-26)**

A central problem in tilings of compact surfaces is the classification of edge-to-edge tilings of the sphere by congruent polygons. It is easy to see that the polygon can only be triangle, quadrilateral, or pentagon. The classification of triangular tilings was started by Sommerville in 1924, and completed by Ueno and Agaoka in 2002.

In 2013, Gao, Shi and Yan classified edge-to-edge tilings of the sphere by 12 pentagons, which was the first classification beyond triangular tilings. Now we have completed the whole classification, i.e., quadrilateral and pentagonal tilings. The tilings are generally of two types: Platonic type and earth map type. There are also several sporadic quadrilateral tilings.

**全等多边形的拼球 - ****Min YAN 严民，香港科技大学 (2023-06-26)**

紧致曲面拼图的一个中心问题是用一个多边形的全等拷贝拼球。在边对边的拼球里，这个多边形只能是三边，四边或五边形。在1924年，Sommerville开始了三边形拼球的研究。在2002年，阿賀岡芳夫和上野裕佳子完成了三边形拼球的分类。

高鸿灏、施楠和我在2013年给出了第一个非三边形拼球的分类结果，即12个全等五边形的拼球。现在所有四边形和五边形拼球都已经完全分类。我们发现主要有两类拼球：柏拉图类，地图类。此外还有几个孤立的四边形拼球。

**Stir fry Homeo_+(S^1) representations from pseudo-Anosov flows - ****Ying HU, University of Nebraska (2023-06-16)**

A total linear order on a non-trivial group G is a left-order if it’s invariant under group left-multiplication. A result of Boyer, Rolfsen and Wiest shows that a $3$-manifold group has a left-order if and only if it admits a non-trivial representation into Homeo_+(S^1) with zero Euler class. Foliations, laminations and flows on $3$-manifolds often give rise to natural non-trivial Homeo_+(S^1)-representations of the fundamental groups, which have proven to be extremely useful in studying the left-orderability of $3$-manifold groups.

In this talk, we will present a recipe of stir frying these Homeo_+(S^1)-representations. Our operation generalizes a previously known ``flipping'' operation introduced by Calegari and Dunfield. As a consequence, we constructed a surprisingly large number of new Homeo_+(S^1)-representations of the link groups. We then use these newly obtained representations to prove the left-orderablity of cyclic branched covers of links associated with any epimorphisms to Z_n. This is joint work with Steve Boyer and Cameron Gordon.

**Watch**

**An infinity-structure in Morse theory - ****Shanzhong SUN 孙善忠, Capital Normal University (2023-06-13)**

In the light of Morse homology initiated by Witten and Floer, we construct two infinity-categories. One comes out of the Morse-Samle pairs and their higher homotopies, and the other strict one concerns the chain complexes of the Morse functions. Based on the boundary structures of the compactified moduli space of gradient flow lines of Morse functions with parameters, we also build up a weak infinity-functor between them. The talk is based on the joint work with with Chenxi WANG.

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**On A Theorem Of Furstenberg - ****Kasra RAFI, University of Toronto (2023-06-09)**

A theorem of Furstenberg from 1967 states that if Gamma is a lattice in a semisimple Lie group G, then there exists a measure on Gamma with finite first moment such that the corresponding harmonic measure on the Furstenberg boundary is absolutely continuous. We will discuss generalizations of this theorem in the setting of the Mapping class group and Gromov hyperbolic groups.

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**On Newton polytopes from the Legendrian dga - ****Orsola CAPOVILLA-SEARLE, UC Davis (2023-06-06)**

In joint work with Roger Casals, we provide a new application of Newton polytopes to the classification of Lagrangian fillings of Legendrian submanifolds in the standard contact R^(2n+1). In particular, we show that Newton polytopes can be used to distinguish infinitely many distinct Lagrangian fillings of Legendrian links in the standard contact R^3 and higher dimensional Legendrian spheres in the standard contact R^(2n+1) sphere up to Hamiltonian isotopy. We also show that there exist Legendrian links with infinitely many exact non-orientable Lagrangian fillings.

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**Contact Hamiltonian Floer homology and its applications - ****Jun ZHANG 张俊, USTC 中科大 (2023-05-29)**

In this talk, we will discuss a Floer-theoretic approach to studying Hamiltonian dynamics on contact manifolds, called contact Hamiltonian Floer homology. On the one hand, it provides a characterization of the rigidity of positive loops in the contactomorphism group which is a peculiar object often investigated in contact geometry. Mysteriously, our characterization highly relates to certain exotic behavior of the Floer continuation map in this setting. On the other hand, from an algebraic perspective, it admits a persistence module type upgrade (we call a gapped module), where one can read off numerical data from the associated barcode. These data help to define contact spectral invariant, contact boundary depth, etc., which also initiates the discovery of novel rigidity phenomenon of subsets in a contact manifold. This talk is based on joint work with Igor Uljarević.

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**Notes**

**Minimal length of nonsimple geodesics in hyperbolic surfaces - ****Wujie SHEN 申武杰, YMSC (2023-05-26)**

The report contains the authors' two articles. We proved that the minimal length M_k for all closed geodesics of self-intersection number at least k among all finite-type hyperbolic surfaces has growth rate 2logk, and furthermore, using the same method, we can also compute the exact value of M_k when k>1750.

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**Notes**

**Geometric decompositions of spherical surfaces - ****Guillaume TAHAR, BIMSA (2023-05-19)**

It is well-known that any plane polygon can be decomposed along its diagonals into flat triangles. The analog does not hold for spherical polygons. We prove that any compact surface with constant positive curvature and conical singularities can be decomposed into irreducible components of standard shape, glued along geodesic arcs connecting conical singularities. The irreducible components include not only spherical triangles but also other interesting spherical polygons called half-spherical concave polygons.

As an application, we prove that any spherical surface with a total angle at least (10g−10+5n)2π contains an embedded sphere with a slit.

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**Contact 3-manifolds and contact surgery - ****Youlin LI 李友林, Shanghai Jiaotong (2023-05-16)**

Contact structures on 3-manifolds are given by a hyperplane distribution in the tangent bundle satisfying a condition called "complete non-integrability". Contact structures fall into one of two classes: tight or overtwisted. Ozsvath and Szabo introduced invariants of contact structures using Heegaard Floer homology. In this talk, I will survey some recent results about the tightness and contact invariants of contact 3-manifolds via contact surgery. The talk is based on joint work with Fan Ding and Zhongtao Wu.

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**Notes**

**Commensurabilities among Lattices in PU(1,n) - ****Chenglong YU 余成龙, YMSC (2023-05-09)**

In simple Lie groups, except the series PU(1,n) with n>1, either lattices are all arithmetic, or mathematicians constructed infinitely many nonarithmetic lattices. So far there are only finitely many nonarithmetic lattices constructed for PU(1,2) and PU(1, 3) and no examples for n>3. One important construction is via monodromy of hypergeometric functions. The discreteness and arithmeticity of those groups are classified by Deligne and Mostow. Thurston also obtained similar results via flat conic metrics. However, the classification of those lattices up to conjugation and finite index (commensurability) is not completed. When n=1, it is the commensurabilities of hyperbolic triangles. The cases of n=2 are almost resolved by Deligne-Mostow and Sauter's commensurability pairs, and commensurability invariants by Kappes-Möller and McMullen. Our approach relies on the study of some higher dimensional Calabi-Yau type varieties instead of complex reflection groups. We obtain some relations and commensurability indices for higher n and also give new proofs for existing pairs in n=2.

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**Non-fillable contact structures on spheres and applications - ****Zhengyi ZHOU 周正一, AMSS (2023-04-25)**

Understanding the landscape of contact structures on spheres is a foundamental question in contact topology. In this talk, I will explain the joint work with Bowden, Gironella and Moreno on exotic contact spheres: the construction of tight contact spheres without fillings and exactly fillable contact spheres without Weinstein fillings. I will also discuss the effect of implementing such contact spheres on other contact manifolds through the contact connected sum.

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**Pseudo-Anosovs of interval type - ****Ethan FARBER, Boston College (2023-04-17)**

A pseudo-Anosov (pA) is a homeomorphism of a compact connected surface S that, away from a finite set of points, acts locally as a linear map with one expanding and one contracting eigendirection. Ubiquitous yet mysterious, pAs have fascinated low-dimensional topologists and dynamicists for the past forty years. We show that any pA on the sphere whose associated quadratic differential has at most one zero, admits an invariant train track whose expanding subgraph is an interval. Concretely, such a pA has the dynamics of an interval map. As an application, we recover a uniform lower bound on the entropy of these pAs originally due to Boissy-Lanneau. Time permitting, we will also discuss potential applications to a question of Fried. This is joint work with Karl Winsor.

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**Notes**

**Asymptotics of the relative Reshetikhin-Turaev invariants - ****Ka Ho WONG 黃嘉豪, Texas A&M University (2023-04-10)**

In a series of joint works with Tian Yang, we made a volume conjecture and an asymptotic expansion conjecture for the relative Reshetikhin-Turaev invariants of a closed oriented 3-manifold with a colored framed link inside it. We propose that their asymptotic behavior is related to the volume, the Chern-Simons invariant and the adjoint twisted Reidemeister torsion associated with the hyperbolic cone metric on the manifold with singular locus the link and cone angles determined by the coloring.

In this talk, I will first discuss how our volume conjecture can be understood as an interpolation between the Kashaev-Murakami-Murakami volume conjecture of the colored Jones polynomials and the Chen-Yang volume conjecture of the Reshetikhin-Turaev invariants. Then I will describe how the adjoint twisted Reidemeister torsion shows up in the asymptotic expansion of the invariants. Especially, we find new explicit formulas for the adjoint twisted Reidemeister torsion of the fundamental shadow link complements and of the 3-manifolds obtained by doing hyperbolic Dehn-filling on those link complements. Those formulas cover a very large class of hyperbolic 3-manifolds and appear naturally in the asymptotic expansion of quantum invariants. Finally, I will discuss some recent progress of the asymptotic expansion conjecture of the fundamental shadow link pairs.

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**Notes**

**Symplectic Excision - ****Xiudi TANG 唐修棣, Beijing Institute of Technology (2023-04-04) **

A symplectic excision is a symplectomorphism between a manifold and the complement of a closed subset. We focus on the construction of symplectic excisions by Hamiltonian vector fields and give some criteria on the existence and non-existence of such kinds of excisions. The talk is based on arXiv:2101.03534.

**Real projective structures on Riemann surfaces and hyper-Kähler metrics - ****Sebastian Heller, ****BIMSA (2023-03-21)**

The non-abelian Hodge theory identifies moduli spaces of representations with moduli spaces of Higgs bundles through solutions to Hitchin's selfduality equations. On the one hand, this enables one to relate geometric structures on surfaces with algebraic geometry, and on the other hand, one obtains interesting hyper-Kähler metrics on the solution spaces. In my talk, I will explain how to construct new hyper-Kähler metrics from certain singular solutions to Hitchin's self-duality equations. The main ingredients are graftings of projective structures, twistor spaces, and Deligne's notion of λ-connections.

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**On the existence of holomorphic curves in compact quotients of SL(2,C) - ****Lynn Heller, ****BIMSA (2023-03-14)**

In my talk, I will report on recent joint work with I. Biswas, S. Dumitrescu and S. Heller showing the existence of holomorphic maps from a compact Riemann surface of genus g>1 into a quotient of SL(2,C) modulo a cocompact lattice which is generically injective. This gives an affirmative answer to a question raised by Huckleberry and Winkelmann and by Ghys. The proof uses ideas from harmonic maps into the hyperbolic 3-space, WKB analysis, and the grafting of real projective structures.

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**Loop decomposition of manifolds - ****Ruizhi Huang, ****BIMSA (2023-03-07)**

The classification of manifolds in various categories is a classical problem in topology. It has been widely investigated by applying techniques from geometric topology in the last century. However, the known results tell us very little information about the homotopy of manifolds. In the last ten years, there have been attempts to study the homotopy properties of manifolds by using techniques in unstable homotopy theory. In this talk, we will discuss the loop decomposition method in this topic and review the known results and our recent work.

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**Notes**

**Systolic inequality on Riemannian manifold with bounded Ricci curvature - ****Zhifei Zhu 朱知非, YMSC (2023-02-28)**

In this talk, we show that the length of a shortest closed geodesic on a Riemannian manifold of dimension 4 with diameter D, volume v, and |Ric|<3 can be bounded by a function of v and D. In particular, this function can be explicitly computed if the manifold is Einstein. The proof of this result depends on a structural theorem proven by J. Cheeger and A. Naber. This is joint work with N. Wu.

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**Notes**

**Equivalent curves on surfaces - ****Binbin XU 徐彬斌, Nankai (2022-12-20)**

We consider a closed oriented surface of genus at least 2. To describe curves on it, one natural idea is to choose once for all a collection of curves as a reference system and to hope that any other curve can be determined by its intersection numbers with reference curves. For simple curves, using the work of Dehn and Thurston, it is possible to find such a reference system consisting of finitely many simple curves. The situation becomes more complicated when curves have self-intersections. In particular, for any non negative integer k, it is possible to find a pair of curves having the same intersection number with every curve with k self-intersections. Such a pair of curves are called k-equivalent curves. In this talk, I will discuss the general picture of a pair of k-equivalent curves and the relation between k-equivalence relations for different k's. This is a joint-work with Hugo Parlier.

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**Notes**

**Limit sets for branching random walks on relatively hyperbolic groups - Wenyuan YANG 杨文元 BICMR, PKU (2022-12-13)**

Branching random walks (BRW) on groups consist of two independent processes on the Cayley graphs: branching and movement. Start with a particle on a favorite location of the graph. According to a given offspring distribution, the particles at the time n split into a random set of particles with mean $r \ge 1$, each of which then moves independently with a fixed step distribution to the next locations. It is well-known that if the offspring mean $r$ is less than the spectral radius of the underlying random walk, then BRW is transient: the particles are eventually free on any finite set of locations. The particles trace a random subgraph which accumulates to a random subset called limit set in a boundary of the graph. In this talk, we consider BRW on relatively hyperbolic groups and study the limit set of the trace at the Bowditch and Floyd boundaries. In particular, the Hausdorff dimension of the limit set will be computed. This is based on a joint work with Mathieu Dussaule and Longmin Wang.

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**Notes**

**Counting incompressible surfaces in 3-manifolds - Nathan DUNFIELD, UIUC (2022-12-06)**

Counting embedded curves on a hyperbolic surface as a function of their length has been much studied by Mirzakhani and others. I will discuss analogous questions about counting incompressible surfaces in a hyperbolic 3-manifold, with the key difference that now the surfaces themselves have more intrinsic topology. As there are only finitely many incompressible surfaces of bounded Euler characteristic up to isotopy in a hyperbolic 3-manifold, it makes sense to ask how the number of isotopy classes grows as a function of the Euler characteristic. Using Haken’s normal surface theory and facts about branched surfaces, we can characterize not just the rate of growth but show it is (essentially) a quasi-polynomial. Moreover, our method allows for explicit computations in reasonably complicated examples. This is joint work with Stavros Garoufalidis and Hyam Rubinstein. Based on https://arxiv.org/abs/2007.10053

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**Notes**

**Teichmüller spaces, quadratic differentials, and cluster coordinates-Dylan ALLEGRETTI, YMSC (2022-11-29)**

Abstract: In the late 1980s, Nigel Hitchin and Michael Wolf independently discovered a parametrization of the Teichmüller space of a compact surface by holomorphic quadratic differentials. In this talk, I will describe a generalization of their result. I will explain how, by replacing holomorphic differentials by meromorphic differentials, one is naturally led to consider an object called the enhanced Teichmüller space. The latter is an extension of the classical Teichmüller space which is important in mathematical physics and the theory of cluster algebras.

**Rigidity in contact topology ****- Honghao GAO 高鸿灏, YMSC (2022-11-22)**

Legendrian links play a central role in low dimensional contact topology. A rigid theory uses invariants constructed via algebraic tools to distinguish Legendrian links. The most influential and powerful invariant is the Chekanov-Eliashberg differential graded algebra, which set apart the first non-classical Legendrian pair and stimulated many subsequent developments. The functor of points for the dga forms a moduli space which acquires algebraic structures and can be used to distinguish exact Lagrangian fillings. Such fillings are difficult to construct and to study, whereas the only known complete classification is the unique filling for Legendrian unknot. A folklore belief was that exact Lagrangian fillings might be scarce. In this talk, I will report a joint work with Roger Casals, where we applied techniques from contact topology, microlocal sheaf theory and cluster algebras to find the first examples of Legendrian links with infinitely many Lagrangian fillings.

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**Notes**

**Title: Separating systole for random hyperbolic surfaces of Weil-Petersson model - Yuhao XUE 薛宇皓, YMSC (2022-11-15)**

In this talk, we will discuss the behavior of the separating systole for random hyperbolic surfaces with respect to the Weil-Petersson measure of the moduli space. We show that its length is approximately 2log(g)-4log(log(g)) and it separates out a one-holed torus for generic surfaces. Some other geometric quantities are also considered. This talk is based on joint works with Xin Nie, Hugo Parlier and Yunhui Wu.

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**Notes**

**Title: Random multicurves on surfaces of large genus and random square-tiled surfaces of large genus - Anton ZORICH, University Paris Cité (2022-11-08)**

It is common in mathematics to study decompositions of compound objects into primitive blocks. For example, the Erdos-Kac Theorem describes the decomposition of a random large integer number into prime factors. There are theorems describing the decomposition of a random permutation of a large number of elements into disjoint cycles.

I will present our formula for the asymptotic count of square-tiled surfaces of any fixed genus g tiled with at most N squares as N tends to infinity. This count allows, in particular, to compute Masur-Veech volumes of the moduli spaces of quadratic differentials. A deep large genus asymptotic analysis of this formula performed by Aggarwal and the uniform large genus asymptotics of intersection numbers of psi-classes on the moduli spaces of complex curves proved by Aggarwal allowed us to describe the decomposition of a random square-tiled surface of large genus into maximal horizontal cylinders. Our results imply, in particular, that with a probability which tends to 1, as genus grows, all "corners" of a random square-tiled surface live on the same horizontal and on the same vertical critical leave.

Maryam Mirzakhani has ingeniously computed frequencies of simple closed multi-geodesics of any topological type on a hyperbolic surface. Developing the results of Mirzakhani we give a detailed portrait of a random hyperbolic multi-geodesics (random multicure) on a Riemann surface of large genus.

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**Notes**

**Stabilizations, Satellites, and Exotic Surfaces - Gary GUTH, University of Oregon**

A long standing question in the study of exotic behavior in dimension four is whether exotic behavior is “stable". For example, in thinking about the four-dimensional h-cobordism theorem, Wall proved that simply connected, exotic four-manifolds always become smoothly equivalent after applying a suitable stabilization operation enough times. Similarly, Hosokawa-Kawauchi and Baykur-Sunukjian showed that exotic surfaces become smoothly equivalent after stabilizing the surfaces some number of times. The question remains, how many stabilizations are necessary, and is one always enough? By considering certain satellite operations, we provide a negative answer to this question in the case of exotic surfaces with boundary. (This draws on joint work with Hayden, Kang, and Park).

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**Positive scalar curvature and exotic aspherical manifolds - Jialong DENG 邓嘉龙, YMSC**

Scalar curvature is interesting not only in analysis, geometry and topology but also in physics. For example, the positive mass theorem, which was proved by Schoen and Yau in 1979, is equivalent to the result that the three-dimension torus carries no Riemannian metric with positive scalar curvature (PSC metric). A widely open conjecture says that a closed aspherical manifold does not admit a PSC metric. I will show that the connected sum of a closed manifold and some exotic aspherical manifolds carries no PSC metric. The enlargeable length-structure and some of Prof. Tom Farrell and his coauthors' work will be used in the talk.

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**Notes**

**Existence and non-existence of Z2 harmonic 1-forms - Siqi HE 何思奇, CAS AMSS (2022-10-18)**

Z2 harmonic 1-forms was introduced by Taubes as the boundary appearing in the compactification of the moduli space of flat SL(2,C) connections. Although from gauge theory aspect, Z2 harmonic 1-forms should exist widely, it is highly challenging to explicitly construct examples of them. Besides the curvature condition, there seems to have more obstruction to the existence of Z2 harmonic 1-forms. In this talk, we will discuss a method to construct examples of Z2 harmonic 1-forms using symmetry. Moreover, we will also discuss the connection between Z2 harmonic 1-forms and special Lagrangian geometry and present a non-existence result.

**Watch** (sincere apologies about the lack of audio to this week's video)

**Notes**

**Almost complex manifolds with prescribed Betti numbers - Zhixu SU 苏之栩, University of Washington （2022-10-11）**

The original version of Sullivan's rational surgery realization theorem provides necessary and sufficient conditions for a prescribed rational cohomology ring to be realized by a simply-connected smooth closed manifold. We will present a version of the theorem for almost complex manifolds. It has been shown there exist closed smooth manifolds M^n of Betti number b_i=0 except b_0=b_{n/2}=b_n=1 in certain dimensions n>16, which realize the rational cohomology ring Q[x]/<x>^3 beyond the well-known projective planes of dimension 4, 8, 16. By the obstructions from the signature equation and the Riemann-Roch integrality conditions among Chern numbers, one can show that none of these manifolds with sum of Betti number three in dimension n>4 can admit almost complex structure. More generally, any 4k (k>1) dimensional closed almost complex manifold with Betti number b_i = 0 except i=0,n/2,n must have even signature and even Euler characteristic, one can characterize all the realizable rational cohomology rings by a set of congruence relations among the signature and Euler characteristic.

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**Notes**

**Equivariant Log-concavity and Equivariant Kahler Packages (or: Shadows of Hodge Theory) - Tao GUI, CAS AMSS (2022-09-27)**

This talk aims to advertise a pattern/phenomenon that has emerged in many different mathematical areas during the past decades but is not currently well-understood. I will begin with a broad overview of the Kahler packages (Poincare duality, Hard Lefschetz, and Hodge-Riemann relations) that appear in geometry, algebra, and combinatorics, from the classics of Lefschetz to the recent work of this year's Fields medalist June Huh, in a down-to-earth way. Then I will discuss two new Kahler packages we discovered that are equivariant and have no geometric origin. The equivariant log-concavity in representation theory hints at our discoveries. This talk will be non-technical and accessible to the general audience: nothing will be assumed other than elementary linear algebra. Partly based on joint work with Rui Xiong.

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**Notes**

**A symplectic approach to 3-manifold triangulations and hyperbolic structures - Dan MATHEWS, Monash University （2022-09-20）**

In the 1980s, Neumann and Zagier introduced a symplectic vector space associated to an ideal triangulation of a cusped 3-manifold, such as a knot complement. We give an interpretation for this symplectic structure in terms of the topology of the 3-manifold, via intersections of certain curves on a Heegaard surface. We also give an algorithm to construct curves forming a symplectic basis for this vector space. This approach gives a description of hyperbolic structures on a knot complement via Ptolemy equations, which can be used to calculate the A-polynomial. This talk involves joint work with Jessica Purcell and Joshua Howie.

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**Notes**

**Introduction to discrete curvature notions (and Graph curvature calculator) - Supanat (Phil) KAMTUE, YMSC (2022-09-13)**

In this talk, I will give a brief introduction to discrete curvature notions and their motivations from Riemannian Geometry. To name a few (which arose and became popular in the last 10-20 years), there are Ollivier Ricci curvature, Bakry-Emery curvature, and Entropic Ricci curvature.

I will help you visualize curvature values in small and simple graphs via this interactive graph curvature calculator Graph Curvature (ncl.ac.uk) created by my colleagues from Newcastle upon Tyne. (And of course, you are welcome to try it beforehand). I will also tell you some stories of our discoveries by playing around this app.

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**Notes**

**Hyperbolic surfaces as singular flat surfaces - Aaron FEYNES, IHÉS (2022-06-29)**

Hyperbolic surfaces and flat surfaces look very different, but they're linked by a remarkable correspondence. I'll show you two versions of it: a geometric "collapsing" process that flattens hyperbolic surfaces, and a representation-theoretic "abelianization" process that diagonalizes SL(2,R) local systems.

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**Notes**

**Intersection number and systole on hyperbolic surfaces - Tina TORKAMAN, Harvard University (2022-06-21)**

Let X be a compact hyperbolic surface. We can see that there is a constant C(X) such that the intersection number of the closed geodesics is bounded above by C(X) times the product of their lengths. Consider the optimum constant C(X). In this talk, we describe its asymptotic behavior in terms of systole, the length of a shortest closed geodesic on X.

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**Nielsen realization problem for 3-manifolds - Lei CHEN 陈蕾, University of Maryland (2022-06-10)**

In this talk, I will describe a joint work with Bena Tshishiku on Nielsen Realization problem for 3-manifolds, in particular, about the twist subgroup. The twist subgroup is a normal finite abelian subgroup of the mapping class group of 3-manifold, generated by the sphere twist. The proof mainly uses the geometric sphere theorem/torus theorem and geometrization.

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**Notes**

**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 AMSS (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.