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Speaker: Weinan Lin (Peking University) Date: 2022-03-03 Location: Room 1114, Sciences Building No. 1 Title: Applications of Groebner basis in algebraic topology Abstract: Given a fixed number of generators for a DGA (differential graded algebra), if there are fewer relations among them, the DGA tends to be larger as a vector space. The extreme cases are polynomial DGAs and DGAs with trivial products. To compute the homology efficiently, one may need to use different algorithms depending on the number of relations. In this talk, I will demonstrate several algorithms I use in my computation of some Ext rings in algebraic topology. Most of the algorithms rely on the theory of Groebner bases. I will give an introduction to Groebner bases before the algorithms. |
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Speaker: Yuxuan Yang (Peking University) Date: 2021-11-30 Location: Room 1114, Sciences Building No. 1 Title: Twists knots的colored Jones多项式的计算 Abstract: Colored Jones多项式是体积猜想中的关键一环。Kazuo Habiro曾经用量子群的办法计算出twists knots的colored Jones多项式。我将汇报Gregor Masbaum的工作，他用Skein理论的办法计算出了上述多项式。 |
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Speaker: Renxing Wan () Date: 2021-11-09 Location: Room 1114, Sciences Building No. 1 Title: Uniform Growth in Groups of Exponential Growth Abstract: This talk is based on two surveys about the growth of groups by Grigorchuk and de la Harpe. At first, I will briefly introduce some notions and list known results and open problems. Later, we will focus on some basic examples including free groups, non-elementary hyperbolic groups, some solvable groups, mapping class group and so on. References: [1] R. Grigorchuk, P. de la Harpe. On problems related to growth, entropy, and spectrum in group theory. Journal of Dynamical and Control Systems. Vol. 3, No. I, 1997, 51-89. https://link.springer.com/content/pdf/10.1007/BF02471762.pdf [2] P. de la Harpe. Uniform Growth in Groups of Exponential Growth. Geometriae Dedicata. Vol. 95, 2002, 1–17. https://link.springer.com/content/pdf/10.1023/A:1021273024728.pdf |
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Speaker: Yue Gao (Peking University) Date: 2021-11-02 Location: Room 1114, Sciences Building No. 1 Title: Shape of filling-systole subspace of surface moduli space by comparing with other subspaces Abstract: This talk is about the space $X_g\subset \mathcal{M}_g$ consisting of surfaces with filling systoles and its subset, critical points of the systole function. In the first part, we obtain a surface with distance $\log\log g$ to $X_g$ and in the second part, prove that a generic point on $\mathcal{M}_g$ has distance $\log\log g$ to $X_g$. Therefore we prove that the radius-$r$ neighborhood of $X_g$ is not able to cover the thick part of $\mathcal M_g$ for any fixed $r>0$. In the last two parts, we get critical points with small and large (comparable to diameter of thick part of $\mathcal M_g$) distance respectively. |
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Speaker: Yi Liu (Beijing International Center for Mathematical Research) Date: 2021-10-26 Location: Room 1114, Sciences Building No. 1 Title: On profinite properties of closed hyperbolic 3-manifolds Abstract: In this talk, I'll discuss some recent progress on profinite completions of fundamental groups of closed hyperbolic 3-manifolds. I'll show that any profinite isomorphism between two such groups determines a bijection between the Zariski dense, algebraic PSL_2 representations, up to conjugacy. Then I'll discuss the question as to learn volume and arithmecity through the profinite completion. |
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Speaker: Jianfeng Lin (Tsinghua University) Date: 2021-10-19 Location: Room 1114, Sciences Building No. 1 Title: Exotic phenomina on 4-manifolds that survive a stabilization Abstract: Starting in dimension 4, there is a significant difference between the category of smooth manifolds and the category of topological manifolds. Such phenomina is called the "exotic phoenomia". An important principle discovered by Wall in the 1960s states that all exotic phenomina on 4-manifolds will dissapear after sufficiently many stabilizations (i.e. connected sum with the product of two spheres). Since then, it has been a long-standing open question whether there exists a pair of homeomorphic simply-connected 4-manifold that are not diffeomorphic after one stabilization. Although we are still not able to solve this problem, in this talk we will present a solution of two variantions: (1) There exits a pair of diffeomorphisms on a 4-manifold that are toplogically isotopic but not smothly isotopic even after a stabilization. (2) There exists a pair of properly embedded surfaces in a 4-manifold with boudary which are topologically isotopic but not smoothly isotopic even after a stablization. (based on joint work with Anuhbav Mukherjee). |
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Speaker: Fan Ye (University of Cambridge) Date: 2021-10-12 Location: Room 1114, Sciences Building No. 1 Title: A large surgery formula for instanton Floer homology Abstract: In Heegaard Floer homology, Oszváth-Szabó and Rasmussen introduced a large surgery formula computing $\widehat{HF}(S^3_m(K))$ for any knot $K$ and large integer $m$ by bent complexes from $CFK^-(K)$. In this talk, I'll introduce a similar formula for instanton Floer homology. More precisely, I construct two differentials on the instanton knot homology $KHI(K)$ and use them to compute the framed instanton homology $I^\#(S^3_m(K))$ for any large integer m. As an application, I show that if the coefficients of the Alexander polynomial of $K$ are not $\pm1$, then there exists an irreducible $SU(2)$ representation of the fundamental group of $S^3_r(K))$ for all but finitely many rational $r$. In particular, all hyperbolic alternating knots satisfy this condition. Also by this large surgery formula, I show $KHI(K)=\widehat{HFK}(K)$ for any Berge knot and $I^\#(S^3_r(K))=\widehat{HF}(S^3_r(K))$ for any genus-one alternating knot. This is a joint work with Zhenkun Li. |
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Speaker: Xingshan Cui (Purdue University) Date: 2021-09-28 Location: Room 1114, Sciences Building No. 1 Title: Two algebraic approaches to constructing invariants of 4-manifolds Abstract: We present two algebraic methods of constructing invariants for smooth closed 4-manifolds. The first one is defined as a state-sum model on triangulations of 4-manifolds and the input data are certain tensor categories and more generally certain higher categories. The invariant can be extended to a topological quantum field theory (TQFT) and extends some of the previously known ones such as Dijkgraaf-Witten and Crane-Yetter/Walker-Wang TQFTs. The second is defined by contracting a tensor diagram assigned to trisections of 4-manifolds and the input data are three Hopf algebras satisfying some consistency conditions. It is not known if this invariant has a TQFT extension, though it includes the Crane-Yetter as special cases. Time permitting, we also mention some possible generalizations of these constructions to obtain potentially more powerful invariants. |
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Speaker: Fan Ye (University of Cambridge) Date: 2021-06-01 Location: Room 1114, Sciences Building No. 1 Title: Dimension bounds on sutured instanton homology SHI Abstract: In this talk, I'll provide a upper bound and a lower bound on the dimension of SHI. In the first part, I'll review the constructions of sutured (Heegaard) Floer homology SFH by Juhász, sutured monopole homology SHM and sutured instanton homology SHI by Kronheimer-Mrowka. In the second part, I'll show dim $KHI$ is less or equal to dim $SFC$, where $SFC$ is the chain complex of $SFH$ for some Heegaard diagram. Also, I'll show graded Euler characteristics of SHI and SFH are the same, which provides a lower bound of dim SHI. As an application, I show $KHI=HFK^hat$ for (1,1)-L-space knots (in particular torus knots) and constrained knots (a generalization of 2-bridge knots in lens spaces). This is a joint work with John A. Baldwin and Zhenkun Li. |
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Speaker: Yi Huang (Tsinghua University) Date: 2021-05-18 Location: Room 1114, Sciences Building No. 1 Title: Simple closed curves on surfaces Abstract: The qualitative and quantitative behaviour of simple closed curves on surfaces can reveal a great deal of geometric information about the underlying surface. We look at three theories within this theme, all pertaining to hyperbolic surfaces: Birman and Series's geodesic sparsity theorem, McShane and Rivin's simple length spectrum growth rate asymptotics (as well as later improvements by Mirzakhani), and McShane identities. I hope to give a feel for why these results hold, as well as my input in extending these results to more general types of surfaces structures. |
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Speaker: Weibiao Wang (Peking University) Date: 2020-11-05 Location: Room 1114, Sciences Building No. 1 Title: Extendable maps on surfaces over the 3-sphere and related topics Abstract: A periodic map on a surface is said to be extendable over the 3-sphere if it can be induced by a periodic automorphism of the 3-sphere on some embedded surface. Recently Chao Wang and I classified and realized all those extendable maps on closed surfaces (in smooth category, with orientation-reversing cases included). I will introduce the work and some related topics, such as the classification of periodic maps on surfaces up to conjugacy, embedded surfaces in lens spaces, and so on. |
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Speaker: Zhenkun Li (Massachusetts Institute of Technology) Date: 2020-01-07 Location: Room 1114, Sciences Building No. 1 Title: Some constructions in sutured monopole and instanton Floer homology Abstract: Sutured monopole Floer homology and sutured Instanton Floer homology were introduced by Kronheimer and Mrowka. They are tools to combines techniques from Gauge theory and the topology of 3-manifolds, and has many remarkable consequences, including a new proof of the Property P conjecture. Despite of those important applications, many basic aspects of the theory remains unclear: the functoriality, the gradings, and its relation with the Thurston norms, etc. In this talk, I will present some constructions and arguments which could resolve some of those mysteries. |
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Speaker: Akio Kawauchi (Osaka City University) Date: 2019-12-30 Location: Room 1479, Sciences Building No. 1 Title: Smooth Poincare conjecture Abstract: |
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Speaker: Akio Kawauchi (Osaka City University) Date: 2019-12-26 Location: Room 1114, Sciences Building No. 1 Title: Knotting Probability of a Spatial Arc Abstract: The knotting probability of an arc diagram is defined as the quadruplet of four kinds of finner knotting probabilities which are invariant under a reasonable deformation containing an isomorphism on an arc diagram. Then it is shown that every oriented spatial arc admits three kinds of arc diagrams unique up to isomorphisms determined from the spatial arc and the projection, so that the knotting probability of a spatial arc is defined. The definition of the knotting probability of and arc diagram uses the fact that every arc diagram induces a unique chord diagram representing a ribbon 2-knot. Then the knotting probability of an arc diagram is set to measure how many non-trivial ribbon genus 2 surface-knots occur from the chord diagram induced from the arc diagram. The condition for an arc diagram with the knotting probability 0 and the condition for an arc diagram with the knotting probability 1 are given together with some other properties and some examples. References: 1) A. Kawauchi, Knotting probability of an arc diagram, http://www.sci.osaka-cu.ac.jp/~kawauchi/diagramknottingprobability.pdf 2) A. Kawauchi, Unique diagram of a spatial arc and the knotting probability, http://www.sci.osaka-cu.ac.jp/~kawauchi/arcknottingprobability.pdf |
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Speaker: Jingling Yang (Chinese University of Hong Kong) Date: 2019-12-19 Location: Room 1114, Sciences Building No. 1 Title: Studies of distance one surgeries on lens space $L(p,1)$ and band surgeries on torus knot $T(2,p)$ Abstract: It has been well known that any closed, orientable 3-manifold can be obtained by Dehn surgery on a link in $S^3$. One of the most prominent problems in 3-manifold topology is to list all the possible lens spaces that can be obtained by a Dehn surgery along a knot in $S^3$, which has been solved by Greene. A natural generalization of this problem is to list all the possible lens spaces that can be obtained by a Dehn surgery from other lens spaces. Besides, considering surgeries between lens spaces is also motivated from DNA topology. In this talk, we will discuss distance one surgeries between lens spaces $L(p,1)$ with $p\geqslant 5$ prime and lens spaces $L(n,1)$ for $n\in Z$, correspondingly band surgeries from $T(2,p)$ to $T(2,n)$, by using Heegaard Floer $d$-invariant. This is a joint work with Zhongtao Wu. |
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Speaker: Yang Su (Academy of Mathematics and Systems Science, Chinese Academy of Sciences) Date: 2019-11-28 Location: Room 1114, Sciences Building No. 1 Title: Free cyclic group actions on $(n-1)$-connected $2n$-manifolds Abstract: In this talk I will present our results on the classification of smooth orientation-preserving free actions of the cyclic group Z/m on (n-1)-connected 2n-manifolds. When n=2 a classification up to topological conjugations is given. When n=3 we obtain a complete classification up to smooth conjugations. For n>3 a complete classification is given when the prime factors of m are large. This is a joint work with Jianqiang Yang. |
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Speaker: Zheyan Wan (Tsinghua University) Date: 2019-11-21 Location: Room 1114, Sciences Building No. 1 Title: Computation of cobordism groups with applications in physics Abstract: Adams spectral sequence is a powerful tool for computing homotopy groups of spectra. Cobordism groups are useful in classification of symmetry protected topological states in physics. By the generalized Pontryagin-Thom isomorphism, the cobordism group $\Omega_d^H(X)$ is exactly the homotopy group $\pi_d(MTH\wedge X_+)$ where $MTH$ is the Madsen-Tillmann spectrum of the group $H$, $X_+$ is the disjoint union of the topological space X and a point. In my talk, I will introduce Adams spectral sequence and compute some cobordism groups with applications in physics. This is my joint work with Juven Wang (arXiv: 1812.11967, 1910.14668, ...). |
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Speaker: Hoang Nyugen (Beijing International Center for Mathematical Research) Date: 2019-10-31 Location: Room 1114, Sciences Building No. 1 Title: Distortion of surfaces in graph manifold Abstract: An immersed surface in a 3-manifold is virtually embedded if the immersion lifts to an embedding into a finite sheeted cover of the manifold. Virtual embedding is equivalent to separability of the surface group in the fundamental group of the 3-manifold. In joint work with Crhis Hruska, we prove that the distortion of a horizontal surface is quadratic if the surface is virtually embedded, and is exponential otherwise. |
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Speaker: Tian-Jun Li (University of Minnesota) Date: 2019-06-20 15:10-17:00 Location: Room 1114, Sciences Building No. 1 Title: Symplectic 4-manifolds with concave boundary Abstract: Symplectic geometry and contact geometry are closely related geometries in even and odd dimensions respectively. And there are two types of symplectic manifolds with contact boundary, the convex ones and the concave ones, depending on the direction of the Reeb vector fields along the boundary. Convex symplectic manifolds, including the Stein manifolds, have been extensively studied in the past 30 years. The concave ones are more abundant and flexible than the convex ones. In this talk I will discuss several results/speculations which indicate that the concave ones also deserve a systematic study, at least in dimension 4. This is a joint work with Cheuk Yu Mak, and partly with Koichi Yasui. |
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Speaker: Chao Wang (East China Normal University) Date: 2019-06-06 15:10-17:00 Location: Room 1479, Sciences Building No. 1 Title: Extending periodic automorphisms of surfaces to 3-manifolds Abstract: Let $S_g$ be the closed orientable surface of genus $g$, $G$ be a finite group acting on it, and $M$ be an integer homology 3-sphere. We show that in the orientable category if each element of $G$ is extendable over $M$ with respect to a fixed embedding from $S_g$ to $M$, then $G$ is extendable over some $M'$ which is 1-dominated by $M$. It has several variations and generalizations. For example, $S_g$ can be replaced by a connected compact manifold or a 3-connected graph. We also classify all orientation-preserving periodic automorphisms of $S_g$ that are extendable over the 3-sphere. The corresponding embedding of such an automorphism can always be an unknotted one. This is a jonit work with Yi Ni and Shicheng Wang. |
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Speaker: Peng Shan (Tsinghua University) Date: 2019-05-09 15:10-17:00 Location: Room 1303, Sciences Building No. 1 Title: Center of G1T-modules and cohomology of affine Springer fibers Abstract: Representations of many remarkable objects, such as Lie algebras, Hecke algebras, algebraic groups, etc., has deep relationships with geometry of some algebraic varieties. A manifestation of such relationship is that sometimes we can realise the center of representation categories as cohomology ring of certain algebraic varieties. In this talk, we will survey some important examples in this direction and explain a new example relating center of G1T-modules and cohomology of affine Springer fibre (joint work with Eric Vasserot). |
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Speaker: Ilya Kapovich (Hunter College) Date: 2019-04-25 15:10-17:00 Location: Room 1114, Sciences Building No. 1 Title: Singularity properties of random free group automorphisms and of random trees in the boundary of Outer space Abstract: It is known that, under mild assumptions, for a free group $F$ of finite rank $r>2$, a "random" element $\phi_n\in Out(F)$ obtained after $n$ steps of a random walk on $Out(F)$ is fully irreducible (a free group analog of being pseudo-Anosov), and that an a.e. trajectory of the way converges to a point in the boundary of the Culler-Vogtmann Outer space $CV(F)$. We prove that generically the attracting $\mathbb R$-tree $T_+(\phi_n)$ for such a random fully irreducible $\phi_n$ is trivalent (that is, all branch points of $T_+$ have degree 3) and non-geometric, that is $T_+$ is not the dual tree of any measured foliation of a finite 2-complex. Similarly, for the exit/harmonic measure of the random walk on the boundary $\partial CV(F)$ of the Outer space, we prove that a generic $\mathbb R$-tree $T\in \partial CV(F)$ is trivalent and non-geometric. The talk is based on joint walk with Joseph Maher, Catherine Pfaff and Samuel Taylor. |
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Speaker: Zhenkun Li (Massachusetts Institute of Technology) Date: 2019-04-11 15:10-17:00 Location: Room 1114, Sciences Building No. 1 Title: Gradings, direct systems and a monopole knot Floer homology Abstract: Given a balanced sutured manifold $(M, \gamma)$ and a properly embedded surface $S$ inside $M$, we can construct a $\mathbb{Z}$ (integral) grading on the sutured monopole Floer homology (SHM) of $(M,\gamma)$. This grading enables us to compute SHM in some cases. As an example, I will explain how to compute the SHM of a solid torus with any valid sutures on its boundary. The grading also plays a crucial role in the construction of a minus version of the monopole knot Floer homology (KHM). Given a knot $K$ in a closed oriented 3-manifold $Y$, the KHM is defined to be the direct limit of a direct system introduced by Etnyre, Vela-Vick and Zarev. This direct system is built on a sequence of balanced sutured manifolds, whose underlying manifolds are all the knot complement $Y(K)=Y\setminus N(K)$ but having different sutures on the boundary. In the talk I will also introduce how the grading is used to prove many interesting properties of this KHM. |
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Speaker: Yu Qiu () Date: 2019-04-04 15:10-17:00 Location: Room 1114, Sciences Building No. 1 Title: Braid groups via cluster exchange groupoids Abstract: We construct braid groups from cluster exchange groupoids with application to the study of moduli spaces of meromorphic quadratic differentials on Riemann surfaces and space of stability conditions on the associated Calabi-Yau-3 categories. This is a joint work with Alastair King. |
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Speaker: Binyong Sun () Date: 2019-03-28 15:10-17:00 Location: Room 1114, Sciences Building No. 1 Title: Low degree cohomologies of congruence groups（同余群的低阶上同调） Abstract: The cohomology of an aspherical space equals the cohomology of the corresponding fundamental group. For congruence groups such as GL(n, Z), we determine its low degree cohomologies (with complex coefficients). Our method is analytic and representation theoretic. Basic theory of continuous cohomologies, and Franke's filtration of the space of automorphic forms will be reviewed. This is a joint work with Jian-Shu Li. |
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Speaker: Liangxia Wan (Beijing Jiaotao University) Date: 2019-03-21 15:10-17:00 Location: Room 1114, Sciences Building No. 1 Title: New presentations of a link and virtual link Abstract: In this talk, I will introduce an embedding presentation of a diagram, which is proved to be a unique presentation of a diagram. Let $\cal L$ be a set of all of diagrams, called also links in this paper. An algebraic system $(\cal L, \sim)$ is constructed. In fact, a link in $R^3$ (or $S^3$) is the equivalent class $[L]$ where $L$ is one of its embedding presentations. Based on $(\cal L, \sim)$, Reduction Crossing Algorithm is proposed which is used to reduce the number of crossings in an embedding presentation by introducing a main tool called a pass replacement. For an infinite set of unknots $\cal U$, each $K$ in $\cal U$ can be transformed into the trivial unknot in at most $O(n^c)$ by applying the algorithm where $c$ is a constant, $K\in {\cal U}$ and $n=|V(K)|$. As special consequences, three unknots are unknotted, which are Goeritz's unknot, Thistlethwaite's unknot and Haken's unknot (image courtesy of Cameron Gordon). Moreover, an infinite family of unknots $K_{G_{2k,2l}}\in {\cal U}$ are unknotted in $O(n\log\log n)$ time. In addition, unique presentations of a virtual link, an oriented link and oriented virtual link are introduced respectively. |
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Speaker: Weibiao Wang (Peking University) Date: 2019-03-14 15:10-17:00 Location: Room 1114, Sciences Building No. 1 Title: A family of pseudo-Anosov braids with small dilatation Abstract: There are homomorphisms from braid groups to mapping class groups, hence braids can be classified according to the Nielsen-Thurston classification. In 2006, Eriko Hironaka and Eiko Kin constructed a family of braids and proved they are pseudo-Anosovwith small dilatation. I will introduce this work. |
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Speaker: Zhiqiang Xiao () Date: 2019-03-07 15:10-17:00 Location: Room 1479, Sciences Building No. 1 Title: Gaps in lattices of (para)topological group topologies and cardinal functions Abstract: I'll introduce the study of lattice of topologies over a given set initiated by Birkhoff in the 1930s and its follow-up developments. Then I mainly focus on the lattice of (para)topological group topologies over a group. So I'll give a brief review of bacic definitions, examples and some classical results in topological group theory firstly, then I discuss about gaps in the lattice of (para)topological group topologies over additive group R of real numbers and the properties of gaps preserved by cardinal funtions. This is a joint work with Wei He, Dekui Peng and Mikhail Tkachenko. |
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Speaker: Ciprian Manolescu (University of California at Los Angeles) Date: 2019-02-28 15:10-17:00 Location: Room 1114, Sciences Building No. 1 Title: A sheaf-theoretic model for $SL(2,\mathbb{C})$ Floer homology Abstract: I will explain the construction of a new homology theory for three-manifolds, defined using perverse sheaves on the $SL(2,\mathbb{C})$ character variety. Our invariant is a model for an $SL(2,\mathbb{C})$ version of Floer's instanton homology. I will present a few explicit computations for Brieskorn spheres, and discuss the connection to the Kapustin-Witten equations and Khovanov homology. This is joint work with Mohammed Abouzaid. |
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Speaker: Yunhui Wu (Tsinghua University) Date: 2019-01-02 Location: Room 1303, Sciences Building No. 1 Title: Small eigenvalues of closed Riemann surfaces for large genus Abstract: We study the asymptotic behavior of small eigenvalues of Riemann surfaces for large genus. We show that for any positive integer $k$, as the genus $g$ goes to infinity, the smallest $k$-th eigenvalue of Riemann surfaces in any thick part of moduli space of Riemann surfacess of genus $g$ is uniformly comparable to $\frac{1}{g^2}$ in $g$. This is a joint work with Yuhao Xue. |
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Speaker: Yue Zhang (University of California at Berkeley) Date: 2018-12-25 Location: Room 1114, Sciences Building No. 1 Title: Guts components of sutured decomposition and the Thurston's Norm Abstract: We define a facet surface of a homology class in an irreducible 3-manifold as a maximal collection of Thurston norm minimizing surfaces and the guts components of this homology class as the nontrivial parts in the sutured decomposition of the 3-manifold along a facet surface. We prove that the guts do not depend on the selection of the facet surfaces and are invariant in the same Thurston cone. Moreover, the rank of the restriction map of the second relative homology from the manifold to the guts is exactly the codimension of the Thurston cone, and the guts of different homology classes are related by some sutured decomposition. In this talk, I will try to give an introduction of sutured decomposition and the guts components. Furthermore, I will show some examples in 3-manifolds which should satisfy our results. This is based on joint work with and supervision of Ian Agol. |
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Speaker: Yi Ni (California Institute of Technology) Date: 2018-12-18 Location: Room 1114, Sciences Building No. 1 Title: The realization problem of prism manifolds Abstract: Prism manifolds are spherical 3-manifolds with D-type finite fundamental groups. They can be parametrized by a pair of relatively prime integers $p>1$ and $q$. The realization problem of prism manifolds asks which prism manifolds can be obtained by positive Dehn surgery on a knot in $S^3$. We will discuss the basic idea of the solution of the realization problem. This talk is based on joint work with (subsets of) Ballinger, Hsu, Mackey, Ochse and Vafaee. |
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Speaker: Jianfeng Lin () Date: 2018-12-17 Location: Room 1114, Sciences Building No. 1 Title: The Pin(2)-equivariant Borsuk–Ulam theorem and the Geography Problem of 4-manifolds Abstract: The classical Borsuk-Ulam theorem states that a continuous map from a n-dimensional sphere to m-dimensional sphere which preserves the antipodal Z/2-actions only exists when m is greater than or equal to n. One can ask a similar question, by replacing the antipodal Z/2-action with an action of the Lie group Pin(2). On a seemingly unrelated side, the Geography Problem of 4-manifolds asks which simply connected topological 4-manifolds admits a smooth structure. By the celebrated works of Kirby-Siebenmann, Freedman, Donaldson, Seiberg-Witten and Furuta, there is a surprising connection between these two questions. In this talk, I will: 1. Explain this beautiful connection between the two problems. 2. Present a solution to the Pin(2)-equivariant Borsuk–Ulam problem. 3. State its application to the Geography Problem. In particular, a partial result on the famous 11/8-conjecture. 4. Describe the ideas of our proof, which uses Pin(2)-equivariant stable homotopy theory. This talk is based on a joint work with Mike Hopkins, XiaoLin Danny Shi and Zhouli Xu. No familiarity of homotopy theory or 4-dimensional topology will be assumed. |
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Speaker: Xueqi Wang (Academy of Mathematics and Systems Science, Chinese Academy of Sciences) Date: 2018-12-11 Location: Room 1114, Sciences Building No. 1 Title: Degrees of maps between $S^3$-bundles over $S^5$ Abstract: In this talk, I will compute all possible degrees of maps between $S^3$-bundles over $S^5$. This will provide a correction of an article by Lafont and Neofytidis. |
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Speaker: Hao Liang () Date: 2018-11-27 Location: Room 1114, Sciences Building No. 1 Title: Homomorphisms to 3-manifold groups Abstract: Following Sela's theory of limit groups (of free group), we define and study limit groups of compact 3-manifold groups. We show that the family of compact 3-manifold groups are equationally noetherian. The main application of our result is to give a positive answer to a question of Reid, Wang and Zhou about epimorphism sequence of closed orientable aspherical 3-manifold groups. This is joint work with Daniel Groves and Michael Hull. |
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Speaker: Haibao Duan (Academy of Mathematics and Systems Science, Chinese Academy of Sciences) Date: 2018-11-04 Location: Room 1114, Sciences Building No. 1 Title: Spin characteristic classes and spin geometry. Abstract: |
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Speaker: Ruifeng Qiu (East China Normal University) Date: 2018-11-04 Location: Room 1114, Sciences Building No. 1 Title: Heegaard Splittings: a survey Abstract: |
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Speaker: Yi Liu (Peking University) Date: 2018-10-23 Location: Room 1114, Sciences Building No. 1 Title: On surface subgroup constructions in finite volume hyperbolic 3-manifolds Abstract: |
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Speaker: Shicheng Wang (Peking University) Date: 2018-10-16 Location: Room 1114, Sciences Building No. 1 Title: 始于手性的漫游 （An unguided tour started from chirality） Abstract: We will report an unguided mathematical tour started from research on chirality since 2000 and, attracted by questions around attractors, led to a zigzag path across topology and dynamics, often switched dimensions. People who joined this tour at various stages include Ding Fan, Jiang Boju, Liu Yi, Ni Yi, Pan Jianzhong, Sun Hongbin, Wang Chao, Wang Shicheng, Yao Jiangang, Zheng Hao, Zhou Qing. Conversations with Robert Edwards, Wen Lan, J. Hillmann, S. Kamada and others, added to the twists and turns that made the trip more fun. |
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Speaker: Wenyuan Yang (Peking University) Date: 2018-10-09 Location: Room 1114, Sciences Building No. 1 Title: Counting conjugacy classes in groups Abstract: In this talk, we introduce a class of statistically convex-cocompact groups and count conjugacy classes when a contracting element is present. Our main result is an asymptotic formula for the number of conjugacy classes of all elements and primitive elements. As corollaries, our results hold for relatively hyperbolic groups, CAT(0) groups with a rank-1 element, and certain subgroups of mapping class groups. Another consequence of this formula is that the generating function for conjugacy classes is transcendental. This is joint work with Ilya Gekhtman (U. Toronto). |
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Speaker: Matthias Kreck (Hausdorff Center for Mathematics) Date: 2018-09-25 Location: Room 1114, Sciences Building No. 1 Title: Invertible topological field theories for manifolds with tangential structures and SKK-groups Abstract: This is joint work with Peter Teichner and Stephan Stolz. A topological field theory associates to a closed $(d-1)$-manifold a vector space and any bordism from one of these manifolds to another a homomorphism between these vector spaces. Here the manifolds are equipped with a tangential structure, e.g. a Spin-structure. Such a field theory is called invertible if the vector space is always 1-dimensional and the homomorphism is an isomorphism. An important axiom for topological field theories is the gluing axiom which already smells a bit like an SK-condition, where SK stands for Schneiden and Kleben = Cutting and Pasting. But this is too restrictive, the partition functions are SKK-invariants, where the second K stands for controlled (kontrolliert in German). We show that the invertible field theories are essentially the same as SKK-invariants. This allows explicit computations in many cases. |
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Speaker: Tian Yang () Date: 2018-08-08 Location: Room 1114, Sciences Building No. 1 Title: Fundamental shadow links Abstract: |
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Speaker: Ying Zhang (Soochow University) Date: 2018-08-08 Location: Room 1114, Sciences Building No. 1 Title: 有限集合上拓扑数的同余性质 Abstract: |
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Speaker: Hongbin Sun () Date: 2018-05-24 Location: Room 1114, Sciences Building No. 1 Title: A characterization on separable subgroups of 3-manifold groups Abstract: We give a complete characterization on which finitely generated subgroups of finitely generated 3-manifold groups are separable. Our characterization generalizes Liu's spirality character on $\pi_1$-injective immersed surface subgroups of closed 3-manifold groups. A consequence of our characterization is that, for any compact, orientable, irreducible and boundary-irreducible 3-manifold M with nontrivial torus decomposition, $\pi_1(M)$ is LERF if and only if for any two adjacent pieces in the torus decomposition of M, at least one of them has a boundary component with genus at least 2. |
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Speaker: Xiaoming Du (South China University of Technology) Date: 2018-05-17 14:30-16:30 Location: Room 1114, Sciences Building No. 1 Title: 从复动力系统的角度去解决 Klein 群极限集形状问题 Abstract: Klein 群是双曲三维流形的基本群。它们的极限集往往是球面上的分形图案。这次报告将用复动力系统轨道闭包形状的方法给出一个集合能否作为某些 Klein 群极限集的判别条件。这是 Benoist 与 Hulin 的工作。报告的原视频见 MSRI 2015 模空间上动力系统会议，论文见 Geo. De. 2018。如果还有时间，将会介绍动力系统的轨道闭包形状的更多结果，包括 Mirzakhani 获菲尔兹奖的工作。 |
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Speaker: Daniel Douglas (Trinity College) Date: 2018-04-23 Location: Room 1114, Sciences Building No. 1 Title: Quantum Traces for Fock-Goncharov Coordinates Abstract: We describe work-in-progress generalizing the SL_2 quantum trace map of Bonahon and Wong (2010) to the case of SL_n. The SL_2 quantum trace is a homomorphism from the Kauffman bracket skein algebra of a punctured surface to a certain noncommutative algebra which can be thought of as a quantum Teichmüller space. The construction is modeled on the classical trace of monodromies of hyperbolic structures on surfaces. Our current work focuses on SL_3, where convex projective structures play the central role, as developed by Fock and Goncharov. Another distinction is the appearance of the HOMFLY-PT skein algebra in place of the Kauffman bracket skein algebra. |
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Speaker: Yi Liu (Beijing International Center for Mathematical Research) Date: 2018-04-19 Location: Room 1114, Sciences Building No. 1 Title: Understanding simple closed curves through finite covers of surfaces Abstract: Given any finite cover of a surface, one may ask whether all the lifted simple closed curves span the first (rational) homology of that cover. In this survey talk, I will review some background of this problem. If time permits, I will briefly explain a negative answer for punctured surfaces, completely following the work of J. Malestein and A. Putman [arXiv:1708.06486]. |
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Speaker: Lei Chen (California Institute of Technology) Date: 2018-03-22 Location: Room 1114, Sciences Building No. 1 Title: From point-picking to sections of surface bundles Abstract: Given any n points on a manifold, how can we systematically and continuously find a new point? What if we ask them to be distinct? In this talk, I will try to answer this question in surfaces. Then I will connect this question to sections of surface bundles. The slogan is "there is no center of mass on closed hyperbolic surfaces". |
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Speaker: Sang-Hyun Kim () Date: 2017-12-16 Location: Room 1114, Sciences Building No. 1 Title: Diffeomorphisms group of one-manifolds. Abstract: |
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Speaker: Bin Yu (Tongji University) Date: 2017-12-14 Location: Room 1114, Sciences Building No. 1 Title: Anosov flows on 3-manifolds Abstract: Abstract: In this talk, we will give an overview of qualitative studies of Anosov flows on 3-manifolds. In particular, we will focus on some progresses about two fundamental questions on this topic: 1. deciding which 3-manifolds admit Anosov flows; 2. classifying Anosov flows on a given 3-manifold. Some open questions associated to these two fundamental questions will also be introduced. |
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Speaker: Shengkui Ye (Xi'an Jiaotong-Liverpool University) Date: 2017-11-23 Location: Room 1114, Sciences Building No. 1 Title: Partial Euler characteristic, normal generations and the stable D(2) problem Abstract: |
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Speaker: Zhe Sun () Date: 2017-10-26 Location: Room 1479, Sciences Building No. 1 Title: Deforming $PSL(n;\mathbb{R})$ Hitchin component, Goldman symplectic form Abstract: (This is joint work with Anna Wienhard and Tengren Zhang.) Let $S$ be a closed, connected, oriented surface of genus at least $2$. It is well-known that on Teichmuller space, any point can deform to another via Thurston's earthquake flow, and the length functions along a pants decomposition of $S$ is a maximal family of Poisson commuting Hamiltonian functions. We prove that any ideal triangulation and any bridge system on S determine a symplectic trivialization (with respect to the Goldman symplectic form) of the tangent bundle of the $PSL(n,\mathbb{R})$ Hitchin component. One can then consider the parallel flows with respect to the flat structure given by this trivialization. We give a geometric description of all such flows in terms of explicit deformations of the associated Frenet curves, and prove that all such flows are Hamiltonian. Applying this to a particular ideal triangulation allows us compute the Goldman symplectic pairing explicitly via labellings on S, thus we can compute the Hamiltonian functions of these Hamiltonian flows explicitly. As a consequence, we find a global Darboux coordinates on $PSL(n,\mathbb{R})$ Hitchin component. |
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Speaker: Xuezhi Zhao (Captial Normal University) Date: 2017-06-27 Location: Room 1479, Sciences Building No. 1 Title: Mapping degrees between spherical $3$-manifolds Abstract: Let $D(M,N)$ be the set of integers that can be realized as the degree of a map between two closed connected orientable manifolds $M$ and $N$ of the same dimension. We determine the set $D(M,N)$ where $M$ and $N$ are closed $3$-manifolds with $S^3$-geometry. |
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Speaker: Feng Luo (Rutgers University) Date: 2017-06-20 Location: Room 1114, Sciences Building No. 1 Title: Discrete uniformization of polyhedral surfaces and its convergence Abstract: We discuss some of the recent work on discrete conformal geometry of polyhedral surfaces. The relationship among discrete conformal geometry, the work of Thurston and Alexandrov on convex surfaces in hyperbolic 3-space, and the Koebe circle domain conjecture will be addressed. We also show that the discrete uniformization maps converge to the conformal maps for disks and tori. This is a joint work with D. Gu, J. Sun, and T. Wu. |
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Speaker: Keiko Kawamuro () Date: 2017-06-06 Location: Room 1114, Sciences Building No. 1 Title: Bennequin-Eliashberg inequality and quasipositive knots and links Abstract: Quasipositive (QP) knots and links in $S^3$ form an important class. Rudolph and Boileau-Orevkov showed that a knot is QP if and only if it is the intersection of an algebraic curve in $\mathbb{C}^2$ and $S^3$. QP links form a monoid of the braid group. Detection of QP and strongly QP knots has been actively studied. It has been questioned by Etnyre, Hedden, Rudolph and Van Horn-Morris whether sharpness of the Bennequin inequality is equivalent to strongly quasipositive. In this talk, I generalize this equivalence to knots and links in general 3-manifolds and give results that support the truth of the equivalence. This is joint work with Tetsuya Ito. |
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Speaker: Yang Su (Academy of Mathematics and Systems Science, Chinese Academy of Sciences) Date: 2017-05-09 Location: Room 1114, Sciences Building No. 1 Title: Mapping class group of spin homology $\mathbb{C}P^3$ Abstract: 光滑流形的映射类群是与流形的几何拓扑有紧密联系的代数对象，它反映了相关几何对象的基本对称性。曲面的映射类群在低维拓扑和几何中具有重要的意义。高维流形的映射类群的计算通常是比较困难的问题，目前只有少数几个例子。在这个报告中我将介绍我和M.Kreck最近合作计算的一族单连通6维流形的映射类群。这些6维流形具有和$\mathbb{C}P^3$相同的同调群。这是我们理解3维复超曲面的映射类群的第一步。 |
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Speaker: Ruifeng Qiu (East China Normal University) Date: 2017-04-23 Location: Room 1114, Sciences Building No. 1 Title: Unknotting number one knots are prime. Abstract: |
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Speaker: Yi Liu (Peking University) Date: 2017-04-18 Location: Room 1114, Sciences Building No. 1 Title: Volume of representation and mapping degree Abstract: |
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Speaker: Thomas Koberda (University of Virginia) Date: 2017-03-07 Location: Room 1114, Sciences Building No. 1 Title: Square roots of Thompson's group F Abstract: I will discuss square roots of Thompson's group F, which are certain two-generator subgroups of the homeomorphism group of the interval, the squares of which generate a copy of Thompson's group F. We prove that these groups may contain nonabelian free groups, they can fail to be smoothable, and can fail to be finitely presented. This represents joint work with Y. Lodha. |
last update: 2022-03-03 12:23:18