Moscow-Beijing topology seminar

Speaker:Vasily Manturov, Wan Zheyan...
Time: Wed 15:20-17:00,2019-7-24 ~ 12-25
Venue:Jing Zhai 112

Speaker Introduction

2019-10-30

Title: Stringc structures and modular invariants

Speaker: 黄瑞芝 Huang Ruizhi (中科院)

Abstract: Spin structure and its higher analogies play important roles in index theory and mathematical physics. In particular, Witten genera for String manifolds have nice geometric implications. As a generalization of the work of Chen-Han-Zhang (2011), we introduce the general Stringc structures based on the algebraic topology of Spinc groups. It turns out that there are infinitely many distinct universal Stringc structures indexed by the infinite cyclic group. Furthermore, we can also construct a family of the so-called generalized Witten genera for Spinc manifolds, the geometric implications of which can be exploited in the presence of Stringc structures. As in the un-twisted case studied by Witten, Liu, etc, in our context there are also integrality, modularity, and vanishing theorems for effective non-abelian group actions. We will also give some applications.

This a joint work with Haibao Duan and Fei Han.

 

2019-10-23

Title: Classification of Links with Khovanov Homology of Minimal Rank

Speaker: 谢羿 Xie Yi(北京大学)

Abstract: Khovanov homology is a link invariant which categorifies the Jones polynomial. It is related to different Floer theories (Heegaard Floer, monopole Floer and instanton Floer) by spectral sequences. It is also known that Khovanov homology detects the unknot, trefoils, unlinks and Hopf links. In this talk, I will give a brief introduction to Khovanov homology and use the instanton Floer homology to prove that links with Khovanov homology of minimal rank must be iterated connected sums and disjoint unions of Hopf links and unknots. This is joint work with Boyu Zhang.

 

2019-10-16

Title: Spin generalization of Dijkraaf-Witten TQFTs

Speaker: Pavel Putrov (ICTP)

Abstract: In my talk I will consider a family of spin topological quantum field theories (spin-TQFTs) that can be considered as spin-version of Dijkgraaf-Witten TQFTs. Although relatively simple, such spin-TQFTs provide non-trivial invariants of (higher-dimensional) links and manifolds, and provide examples of categorification of such quantum invariants.

 

2019-10-9

Title: Virtual Knots and Links and Perfect Matchings of Trivalent Graphs

Speaker: Louis H. Kauffman (UIC and NSU)

Abstract: In this talk we discuss a mapping from Graphenes (oriented perfect matching structures on trivalent graphs with cyclic orders at their vertices) to Virtual Knots and Links. We show how, with an appropriate set of moves on Graphenes, our mapping K: Graphenes —> Virtual Knots and Links is an equivalence of categories. This means that we can define new invariants of graphenes by using invariants of virtual knots and links, including Khovanov homology for graphenes. The equivalence K allows us to explore problems about graphs, such as coloring problems and flow problems, in terms of knot theory. The talk will introduce many examples of this correspondence and discuss how the classical coloring problems for graphs are illuminated by the topology of virtual knot theory.

 

2019-9-25

Title: Simplicial structures in braid/knot theory and data analytics

Speaker: 吴杰Wu Jie

Abstract: In this talk, we will explain simplicial technique on braids and links as well as its applications in data science.

 

2019-9-18

Title: Computation of Spin cobordism groups

Speaker: 万喆彦 Wan Zheyan (YMSC)

Abstract: Adams spectral sequence is a powerful tool for computing homotopy groups of spectra. In particular, it was used for computing homotopy groups of sphere spectrum, which are stable homotopy groups of spheres. By the generalized Pontryagin-Thom isomorphism, the Spin cobordism group \Omega_d^{Spin}(X) is exactly the homotopy group \pi_d(MSpin\wedge X_+) where MSpin is the Thom spectrum, X_+ is the disjoint union of X and a point. In my talk, I will introduce spectra, Adams spectral sequence and compute the Spin cobordism groups for a special topological space X. These are contained in my joint work with Juven Wang (arXiv: 1812.11967).

 

2019-9-11

Title: Word problem in certain G_n^k groups

Speaker: Denis Fedoseev (莫斯科国立大学)

Abstract: In the present talk we discuss the word and, to lesser extent, the conjugacy problems in certain groups G_n^k, which were introduced by V. Manturov.

We prove that the word problem is algorithmically solvable in the groups G_4^3 and G_5^4.

The talk is based on the joint work with V. Manturov and A. Karpov.

 

2019-9-4

Title: Coxeter arrangements in three dimensions

Speaker: 王军 Wang Jun (首都师范大学)

Abstract: Let $\mathcal{A}$ be a finite real linear hyperplane arrangement in three dimensions. Suppose further that all the regions of $\mathcal{A}$ are isometric. The open quesion asked by C.Klivans and E.Swartz in 2001 is that does there exist a real central hyperplane arrangement with all regions isometric that is not a Coxeter arrangement.

In 2016, R.Ehrenborg, C.Klivans and N.Reading proved that in three dimensions, $\mathcal{A}$ is necessarily a Coxeter arrangement.  As it is well known that the regions of a Coxeter arrangement are isometric, this characterizes three-dimensional Coxeter arrangements precisely as those arrangements with isometric regions. It is an open question whether this suffices to characterize Coxeter arrangements in higher dimensions.

 In this talk, we will intruduce Coxeter arrangement and the proof of R.Ehrenborg, C.Klivans and N.Reading in three dimensions.

 

2019-8-28

Title: Pictures instead of coefficients: the label bracket

Speaker: Vassily Manturov (莫斯科国立技术大学)

Abstract: We shall consider skein-relations where instead of coefficients we draw small pictures.

This way, starting from the Kauffman bracket formalism, we get to a knot invariant (valued in pictures with modulo relations) which dominates

not only the Kauffman bracket but also the Kuperberg bracket, the HOMFLY polynomial, the arrow polynomial and the Kuperberg picture-valued invariant for virtual knots and knotoids.

This is a joint work with Alyona Akimova and Louis Kauffman.

https://arxiv.org/abs/1907.06502

 

2019-8-21

Title: Knots with identical Khovanov homology

Speaker: 柏升 Bai Sheng(北京化工大学)

Abstract: This is Liam Watson's paper in  Algebraic & Geometric Topology 7 (2007) 1389–1407. He gives a recipe for constructing families of distinct knots that have identical Khovanov homology and give examples of pairs of prime knots, as well as infinite families, with this property.

 

2019-8-14

Title: A Discrete Morse Theory for Hypergraphs

Speaker: 任世全Ren Shiquan(清华大学)

Abstract: A hypergraph can be obtained from a simplicial complex by deleting some non-maximal simplices. By [A.D. Parks and S.L. Lipscomb, Homology and hypergraph acyclicity: a combinatorial invariant for hypergraphs. Naval Surface Warfare Center, 1991], a hypergraph gives an associated simplicial complex. By [S. Bressan, J. Li, S. Ren and J. Wu, The embedded homology of hypergraphs and applications. Asian J. Math. 23 (3) (2019), 479-500], the embedded homology of a hypergraph is the homology of the infimum chain complex, or equivalently, the homology of the supremum chain complex. In this paper, we generalize the discrete Morse theory for simplicial complexes by R. Forman [R. Forman, Morse theory for cell complexes. Adv. Math. 134 (1) (1998), 90-145], [R. Forman, A user’s guide to discrete Morse theory. Séminaire Lotharingien de Combinatoire 48 Article B48c, 2002], [R. Forman, Discrete Morse theory and the cohomology ring. Trans. Amer. Math. Soc. 354 (12) (2002), 5063-5085] and give a discrete Morse theory for hypergraphs. We use the critical simplices of the associated simplicial complex to construct a sub-chain complex of the infimum chain complex and a sub-chain complex of the supremum chain complex, then prove that the embedded homology of a hypergraph is isomorphic to the homology of the constructed chain complexes. Moreover, we define discrete Morse functions on hypergraphs and compute the embedded homology in terms of the critical hyperedges. As by-products, we derive some Morse inequalities and collapse results for hypergraphs.

 

2019-8-7

Title: Pre-image classes and an application in knot theory

Speaker: 赵学志Zhao Xuezhi(首都师范大学)

Abstract: Let $f: X\to Y$ be a map and $B$ be a non-empty closed subset of $Y$. We consider the pre-image $f^{-1}(B)$ according to Nielsen fixed theory. Can we give a lower bound for the number of components $g^{-1}(B)$, where $g$ is an arbitrary map in the homotopy class of $f$? This is a natural generalization of root theory. We shall apply this theory to give invariants for knots and links.

 

2019-7-31

Title: Mysteries of approximation in $\mathbb{R}^4$

Speaker: N.G.Moshchevitin(莫斯科国立大学)

Abstract: We will discuss some unsolved problems in the theory of Diophantine Approximation.

It turns out that certain questions related to approximation of subspaces of $\mathbb{R}^4$ by rational subspaces remain unclear. We discuss some of them as well as related topics.

 

2019-7-24

Title: Quandle and its applications in knot theory

Speaker: 程志云Cheng Zhiyun(北京师范大学)

Abstract: In this talk I will give a brief introduction to quandle theory. Several applications of quandle in knot theory will be discussed.

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