17:00 -- 18:30 数理科学研究科棟(東京大学駒場キャンパス)
Tea: 16:30 -- 17:00 コモンルーム

Last updated April 14, 2017
河野 俊丈
河澄 響矢
北山 貴裕
逆井 卓也

4月11日 -- 056号室, 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.

4月18日 -- 056号室, 17:00 -- 18:30

野坂 武史 (東京工業大学)


Abstract: われわれは、ミルナー不変量を、群の中心拡大と冪単マグナス展開をもちいて再構成した。 それにより当不変量の図式計算方法を確立した。本講演ではその再構成と計算法を説明し、いくつか例示をする。 また冪零的マグナス展開の性質も紹介したい。本研究は九大の小谷久寿氏との共同研究である。

4月25日 -- 056号室, 17:00 -- 18:30

久野 雄介 (津田塾大学)

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.

5月9日 -- 056号室, 17:00 -- 18:30

諏訪 立雄 (北海道大学)

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.

5月16日 -- 056号室, 17:00 -- 18:30

合田 洋 (東京農工大学)

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.

5月30日 -- 056号室, 17:00 -- 18:30

森藤 孝之 (慶應義塾大学)

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.