Topology

2009年4月 -- 7月

[English] [過去のプログラム]

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

Last updated June 30, 2009
世話係　
河野俊丈
河澄響矢

4月21日 -- 056号室, 16:30 -- 18:00

Ivan Marin (Univ. Paris VII)

Some algebraic aspects of KZ systems

Abstract: Knizhnik-Zamolodchikov (KZ) systems enables one to construct representations of (generalized) braid groups. While this geometric construction is now very well understood, it still brings to attention, or helps constructing, new algebraic objects. In this talk, we will present some of them, including an infinitesimal version of Iwahori-Hecke algebras and a generalization of the Krammer representations of the usual braid groups.

4月28日 -- 056号室, 16:30 -- 18:00

平地健吾 (東京大学大学院数理科学研究科)

The ambient metric in conformal geometry

Abstract: In 1985, Charles Fefferman and Robin Graham gave a method for realizing a conformal manifold of dimension n as a submanifold of a Ricci-flat Lorentz metric on a manifold of dimension n+2, which is now called the ambient space. Using this correspondence, one can construct many examples of conformal invariants and conformally invariant operators. However, if n is even, their construction of the ambient space is obstructed at the jet of order n/2 and thereby the application of the ambient space was limited. In this talk, I'll recall basic ideas of the ambient space and then explain how to avoid the difficulty and go beyond the obstruction. This is a joint work with Robin Graham.

5月12日 -- 056号室, 16:30 -- 18:00

松田能文 (東京大学大学院数理科学研究科)

円周の微分同相群の離散部分群

Abstract: 円周の微分同相群の離散部分群の典型例はフックス群である. この講演では, 円周の向きを保つ実解析的微分同相群の離散部分群であってフックス群の有限被覆と位相共役でないものを構成する.

5月19日 -- 056号室, 16:30 -- 18:00

Mark Hamilton (東京大学大学院数理科学研究科, JSPS)

Geometric quantization of integrable systems

Abstract: The theory of geometric quantization is one way of producing a "quantum system" from a "classical system," and has been studied a great deal over the past several decades. It also has surprising ties to representation theory. However, despite this, there still does not exist a satisfactory theory of quantization for systems with singularities.

Geometric quantization requires the choice of a polarization; when using a real polarization to quantize a regular enough manifold, a result of Sniatycki says that the quantization can be found by counting certain objects, called Bohr-Sommerfeld fibres. However, there are many types of systems to which this result does not apply. One such type is the class of completely integrable systems, which are examples coming from mechanics that have many nice properties, but which are nontheless too singular for Sniatycki's theorem to apply.

In this talk we will explore one approach to the quantization of integrable systems, and show a Sniatycki-type relationship to Bohr-Sommerfeld fibres. However, some surprising features appear, including infinite-dimensional contributions and strong dependence on the polarization.

This is joint work with Eva Miranda.

5月26日 -- 056号室, 16:30 -- 18:00

境圭一 (東京大学大学院数理科学研究科)

配置空間積分と long embedding の空間のコホモロジー

Abstract: 三価グラフに付随する配置空間積分により、(long) 1-knot に対する有限型不変量 (Bott-Taubes, Kohno, ...) や、余次元2の奇数次元 long embedding の不変量 (Bott, Cattaneo-Rossi, Watanabe) などの非自明なコホモロジー類が構成されることが知られています。本講演では、配置空間積分により、三価でないグラフに対応するコホモロジー類や、6次元空間への long 3-embedding に対する Haefliger 不変量の新しい定式化など、さらに多くのコホモロジー類が得られることを、Budney による little balls operad の作用、Roseman-Takase の deform-spinning などと関連付けながら述べたいと思います（一部は渡邉忠之氏との共同研究）。

6月2日 -- 056号室, 16:30 -- 18:00

Alexander Voronov (University of Minnesota)

Graph homology: Koszul duality = Verdier duality

Abstract: Graph cohomology appears in computation of the cohomology of the moduli space of Riemann surfaces and the outer automorphism group of a free group. In the former case, it is graph cohomology of the commutative and Lie types, in the latter it is ribbon graph cohomology, that is to say, graph cohomology of the associative type. The presence of these three basic types of algebraic structures hints at a relation between Koszul duality for operads and Poincare-Lefschetz duality for manifolds. I will show how the more general Verdier duality for certain sheaves on the moduli spaces of graphs associated to Koszul operads corresponds to Koszul duality of operads. This is a joint work with Andrey Lazarev.

6月9日 -- 056号室, 16:30 -- 18:00

五味清紀 (京都大学大学院理学研究科)

A finite-dimensional construction of the Chern character for twisted K-theory

Abstract: Twisted K-theory is a variant of topological K-theory, and is attracting much interest due to applications to physics recently. Usually, twisted K-theory is formulated infinite-dimensionally, and hence known constructions of its Chern character are more or less abstract. The aim of my talk is to explain a purely finite-dimensional construction of the Chern character for twisted K-theory, which allows us to compute examples concretely. The construction is based on twisted version of Furuta's generalized vector bundle, and Quillen's superconnection. This is a joint work with Yuji Terashima.

6月16日 -- 056号室, 16:30 -- 18:00

佐藤正寿 (東京大学大学院数理科学研究科)

レベル2写像類群のアーベル化

Abstract: レベルつき写像類群は曲面の写像類群の指数有限部分群である. 特にdが3以上において, そのアーベル化はレベル構造をもつ非特異代数曲線のモジュライ空間の整係数1次ホモロジー群と一致する. レベルd写像類群のアーベル化を, dが奇数もしくは2について完全に, 一般の偶数dについてもほぼ決定できたので, これについて紹介する.

6月23日 -- 056号室, 16:30 -- 18:00

久野雄介 (東京大学大学院数理科学研究科)

射影多様体に対するMeyer函数とその応用

Abstract: Meyer函数は、曲面上の曲面束の符号数に関連する一種の二次不変量である。この講演では、非特異射影多様体に対してMeyer函数が一意的に存在することを示す。この函数は、ある開代数多様体の基本群の上の類函数となっている。ファイバー構造を持つ4次元多様体の局所符号数への応用についても述べる予定である。

6月30日 -- 056号室, 16:30 -- 18:00

北山貴裕 (東京大学大学院数理科学研究科)

Torsion volume forms and twisted Alexander functions on character varieties of knots

Abstract: Using non-acyclic Reidemeister torsion, we can canonically construct a complex volume form on each component of the lowest dimension of the $SL_2(\mathbb{C})$-character variety of a link group. This volume form enjoys a certain compatibility with the following natural transformations on the variety. Two of them are involutions which come from the algebraic structure of $SL_2(\mathbb{C})$ and the other is the action by the outer automorphism group of the link group. Moreover, in the case of knots these results deduce a kind of symmetry of the $SU_2$-twisted Alexander functions which are globally described via the volume form.

7月14日 -- 056号室, 17:00 -- 18:00

作間誠 (広島大学)

The Cannon-Thurston maps and the canonical decompositions of punctured-torus bundles over the circle.

Abstract: To each once-punctured-torus bundle over the circle with pseudo-Anosov monodromy, there are associated two tessellations of the complex plane: one is the triangulation of a horosphere induced by the canonical decomposition into ideal tetrahedra, and the other is a fractal tessellation given by the Cannon-Thurston map of the fiber group. In this talk, I will explain the relation between these two tessellations (joint work with Warren Dicks). I will also explain the relation of the fractal tessellation and the "circle chains" of double cusp groups converging to the fiber group (joint work with Caroline Series). If time permits, I would like to discuss possible generalization of these results to higher-genus punctured surface bundles.