Tuesday Seminar on Topology

17:00 -- 18:30 Graduate School of Mathematical Sciences, The University of Tokyo
Tea: 16:30 -- 17:00 Common Room

Last updated September 4, 2017
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Toshitake Kohno
Nariya Kawazumi
Takahiro Kitayama
Takuya Sakasai

April 11 -- Room 056, 17:00 -- 18:30

Alexander Voronov (University of Minnesota)

Homotopy Lie algebroids

Abstract: A well-known result of A. Vaintrob [Vai97] characterizes Lie algebroids and their morphisms in terms of homological vector fields on supermanifolds. We give an interpretation of Lie bialgebroids and their morphisms in terms of odd symplectic dg-manifolds, building on the approach of D. Roytenberg[Roy99]. This extends naturally to the homotopy Lie case and leads to the notion of L-bialgebroids and L-morphisms between them.

April 18 -- Room 056, 17:00 -- 18:30

Takefumi Nosaka (Tokyo institute of Technology)

Milnor invariants via unipotent Magnus embeddings

Abstract: We reconfigured the Milnor invariant, in terms of central group extensions and unipotent Magnus embeddings, and develop a diagrammatic computation of the invariant. In this talk, we explain the reconfiguration and the computation with mentioning some examples. I also introduce some properties of the unipotent embeddings. This is a joint work with Hisatoshi Kodani.

April 25 -- Room 056, 17:00 -- 18:30

Yusuke Kuno (Tsuda University)

Formality of the Goldman-Turaev Lie bialgebra and the Kashiwara-Vergne problem in positive genus

Abstract: This talk is based on a joint work with A. Alekseev, N. Kawazumi and F. Naef. Given a compact oriented surface with non-empty boundary and a framing of the surface, one can define the Lie bracket (Goldman bracket) and the Lie cobracket (Turaev bracket) on the vector space spanned by free homotopy classes of loops on the surface. These maps are of degree minus two with respect to a certain filtration. Then one can ask the formality of this Lie bialgebra: is the Goldman-Turaev Lie bialgebra isomorphic to its associated graded?
For surfaces of genus zero, we showed that this question is closely related to the Kashiwara-Vergne (KV) problem in Lie theory (arXiv:1703.05813). A similar result was obtained by G. Massuyeau by using the Kontsevich integral.
Our new topological interpretation of the classical KV problem leads us to introduce a generalization of the KV problem in connection with the formality of the Goldman-Turaev Lie bialgebra for surfaces of positive genus. We will discuss the existence and uniqueness of solutions to the generalized KV problem.

May 9 -- Room 056, 17:00 -- 18:30

Tatsuo Suwa (Hokkaido University)

Local and global coincidence homology classes

Abstract: We consider two differentiable maps between two oriented manifolds. In the case the manifolds are compact with the same dimension and the coincidence points are isolated, there is the Lefschetz coincidence point formula, which generalizes his fixed point formula. In this talk we discuss the case where the dimensions of the manifolds may possible be different so that the coincidence points are not isolated in general. In fact it seems that Lefschetz himself already considered this case (cf. [4]).
We introduce the local and global coincidence homology classes and state a general coincidence point theorem. We then give some explicit expressions for the local class. We also take up the case of several maps as considered in [1] and perform similar tasks. These all together lead to a generalization of the aforementioned Lefschetz formula. The key ingredients are the Alexander duality in combinatorial topology, intersection theory with maps and the Thom class in Čech-de Rham cohomology. The contents of the talk are in [2] and [3].

[1] C. Biasi, A.K.M. Libardi and T.F.M. Monis, The Lefschetz coincidence class of p maps, Forum Math. 27 (2015), 1717-1728.
[2] C. Bisi, F. Bracci, T. Izawa and T. Suwa, Localized intersection of currents and the Lefschetz coincidence point theorem, Annali di Mat. Pura ed Applicata 195 (2016), 601-621.
[3] J.-P. Brasselet and T. Suwa, Local and global coincidence homology classes, arXiv:1612.02105.
[4] N.E. Steenrod, The work and influence of Professor Lefschetz in algebraic topology, Algebraic Geometry and Topology: A Symposium in Honor of Solomon Lefschetz, Princeton Univ. Press 1957, 24-43.

May 16 -- Room 056, 17:00 -- 18:30

Hiroshi Goda (Tokyo University of Agriculture and Technology)

Twisted Alexander invariants and Hyperbolic volume of knots

Abstract: In [1], Müller investigated the asymptotics of the Ray-Singer analytic torsion of hyperbolic 3-manifolds, and then Menal-Ferrer and Porti [2] have obtained a formula on the volume of a hyperbolic 3-manifold using the Higher-dimensional Reidemeister torsion.
On the other hand, Yoshikazu Yamaguchi has shown that a relationship between the twisted Alexander polynomial and the Reidemeister torsion associated with the adjoint representation of the holonomy representation of a hyperbolic 3-manifold in his thesis [3].
In this talk, we observe that Yamaguchi's idea is applicable to the Higher-dimensional Reidemeister torsion, then we give a volume formula of a hyperbolic knot using the twisted Alexander polynomial.

[1] Müller, W., The asymptotics of the Ray-Singer analytic torsion of hyperbolic 3-manifolds, Metric and differential geometry, 317--352, Progr. Math., 297, Birkhäuser/Springer, Basel, 2012.
[2] Menal-Ferrer, P. and Porti, J., Higher-dimensional Reidemeister torsion invariants for cusped hyperbolic 3-manifolds. J. Topol., 7 (2014), no. 1, 69--119.
[3] Yamaguchi, Y., On the non-acyclic Reidemeister torsion for knots, Dissertation at the University of Tokyo, 2007.

May 23 -- Lecture Hall, 17:00 -- 18:30

Richard Hain (Duke University)

Johnson homomorphisms, stable and unstable

Abstract: In this talk I will recall how motivic structures (Hodge and/or Galois) on the relative completions of mapping class groups yield non-trivial information about Johnson homomorphisms. I will explain how these motivic structures can be pasted, and why I believe that the genus 1 case is of fundamental importance in studying the higher genus (stable) case.

May 30 -- Room 056, 17:00 -- 18:30

Takayuki Morifuji (Keio University)

On a conjecture of Dunfield, Friedl and Jackson for hyperbolic knots

Abstract: The hyperbolic torsion polynomial is defined to be the twisted Alexander polynomial associated to the holonomy representation of a hyperbolic knot. Dunfield, Friedl and Jackson conjecture that the hyperbolic torsion polynomial determines the genus and fiberedness of a hyperbolic knot. In this talk we will survey recent results on the conjecture and explain its generalization to hyperbolic links.

June 6 -- Room 056, 17:00 -- 18:30

Shunsuke Tsuji (The University of Tokyo)

A formula for the action of Dehn twists on the HOMFLY-PT type skein algebra and its application

Abstract: We give an explicit formula for the action of the Dehn twist along a simple closed curve of a surface on the completed HOMFLY-PT type skein modules of the surface in terms of the action of the completed HOMFLY-PT type skein algebra of the surface. As an application, using this formula, we construct an invariant for an integral homology 3-sphere which is an element of Q[ρ] [[h]].

June 13 -- Room 056, 17:00 -- 18:30

Noboru Ogawa (Tokai University)

Local criteria for non-embeddability of Levi-flat manifolds

Abstract: In this talk, we consider the Levi-flat embedding problem. Barrett showed that a smooth Reeb foliation on S^3 cannot be realized as a Levi-flat hypersurface in any complex surfaces. To do this, he focused the relationship between the holonomy along the compact leaf and a system of its defining functions. We will show a new criterion for non-embeddability of Levi-flat manifolds. Our result is a higher dimensional analogue of Barrett's theorem. In particular, this enables us to weaken the compactness assumption. For this purpose, we pose a partial generalization of Ueda theory on the analytic neighborhood structure of complex hypersurfaces. This talk is based on a joint work with Takayuki Koike (Kyoto University).

June 20 -- Room 056, 17:00 -- 18:30

Anh Tran (The University of Texas at Dallas)

Introduction to the AJ conjecture

Abstract: The AJ conjecture was proposed by Garoufalidis about 15 years ago. It predicts a strong connection between two important knot invariants derived from very different background, namely the colored Jones function (a quantum invariant) and the A-polynomial (a geometric invariant). The colored Jones function is a sequence of Laurent polynomials which is known to satisfy a linear q-difference equation. The AJ conjecture states that by writing the linear q-difference equation into an operator form and setting q=1, one gets the A-polynomial. In this talk, I will give an introduction to this conjecture.

June 27 [Joint with RIKEN iTHEMS] -- Room 056, 17:00 -- 18:30

Eiko Kin (Osaka University)

Braids and hyperbolic 3-manifolds from simple mixing devices

Abstract: Taffy pullers are devices for pulling candy. One can build braids from the motion of rods for taffy pullers. According to a beautiful article ``A mathematical history of taffy pullers" by Jean-Luc Thiffeault, all taffy pullers (except the first one) give rise to pseudo-Anosov braids. This means that the devices mix candies effectively. Following a study of Thiffeault, I will discuss which pseudo-Anosov braid is realized by taffy pullers. I will explain an interesting connection between braids coming from taffy pullers. I also discuss the hyperbolic mapping tori obtained from taffy pullers. Intriguingly, the two most common taffy pullers give rise to the complements of the the minimally twisted 4-chain link and 5-chain link which are important examples for the study of cusped hyperbolic 3-manifolds with small volumes.

Reference: A mathematical history of taffy pullers, Jean-Luc Thiffeault, https://arxiv.org/pdf/1608.00152.pdf

July 4 -- Room 056, 17:00 -- 18:30

Jean-Baptiste Meilhan (Université Grenoble Alpes)

On link-homotopy for knotted surfaces in 4-space

Abstract: The purpose of this talk is to show how combinatorial objects (welded objects, which is a natural quotient of virtual knot theory) can be used to study knotted surfaces in 4-space.
We will first consider the case of 'ribbon' knotted surfaces, which are embedded surfaces bounding immersed 3-manifolds with only ribbon singularities. More precisely, we will consider ribbon knotted annuli ; these objects act naturally on the reduced free group, and we prove, using welded theory, that this action gives a classification up to link-homotopy, that is, up to continuous deformations leaving distinct component disjoint. This in turns implies a classification result for ribbon knotted tori.
Next, we will show how to extend this classification result beyond the ribbon case.
This is based on joint works with B. Audoux, P. Bellingeri and E. Wagner.

July 11 -- Room 056, 17:00 -- 18:30

Celeste Damiani (JSPS, Osaka City University)

Some remarkable quotients of virtual braid groups

Abstract: Virtual braid groups are one of the most famous generalizations of braid groups. We introduce a family of quotients of virtual braid groups, called loop braid groups. These groups have been an object of interest in different domains of mathematics and mathematical physics, and can be found in the literature also by names such as groups of permutation-conjugacy automorphisms, braid- permutation groups, welded braid groups, weakly virtual braid groups, untwisted ring groups, and others. We show that they share with braid groups the property of admitting many different definitions. After that we consider a further family of quotients called unrestricted virtual braids, describe their structure and explore their relations with fused links.