## Moscow-Beijing topology seminar

## 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.