Topology

2010年10月 -- 2011年1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 January 24, 2011
世話係　
河野俊丈
河澄響矢

10月12日 -- 056号室, 16:30 -- 18:00

Andrei Pajitnov (Univ. de Nantes, 東京大学大学院数理科学研究科)

Asymptotics of Morse numbers of finite coverings of manifolds

Abstract: Let X be a closed manifold; denote by m(X) the Morse number of X (that is, the minimal number of critical points of a Morse function on X). Let Y be a finite covering of X of degree d.
In this survey talk we will address the following question posed by M. Gromov: What are the asymptotic properties of m(N) as d goes to infinity?
It turns out that for high-dimensional manifolds with free abelian fundamental group the asymptotics of the number m(N)/d is directly related to the Novikov homology of N. We prove this theorem and discuss related results.

10月19日 -- 002号室, 16:30 -- 18:00

Jinseok Cho (早稲田大学)

Optimistic limits of colored Jones invariants

Abstract: Yokota made a wonderful theory on the optimistic limit of Kashaev invariant of a hyperbolic knot that the limit determines the hyperbolic volume and the Chern-Simons invariant of the knot. Especially, his theory enables us to calculate the volume of a knot combinatorially from its diagram for many cases.
We will briefly discuss Yokota theory, and then move to the optimistic limit of colored Jones invariant. We will explain a parallel version of Yokota theory based on the optimistic limit of colored Jones invariant. Especially, we will show the optimistic limit of colored Jones invariant coincides with that of Kashaev invariant modulo 2\pi^2. This implies the optimistic limit of colored Jones invariant also determines the volume and Chern-Simons invariant of the knot, and probably more information.
This is a joint-work with Jun Murakami of Waseda University.

10月26日 -- 056号室, 17:00 -- 18:00

葉廣和夫 (京都大学数理解析研究所)

Quantum fundamental groups and quantum representation varieties for 3-manifolds

Abstract: We define a refinement of the fundamental groups of 3-manifolds and a generalization of representation variety of the fundamental group of 3-manifolds. We consider the category $H$ whose morphisms are nonnegative integers, where $n$ corresponds to a genus $n$ handlebody equipped with an embedding of a disc into the boundary, and whose morphisms are the isotopy classes of embeddings of handlebodies compatible with the embeddings of the disc into the boundaries. For each 3-manifold $M$ with an embedding of a disc into the boundary, we can construct a contravariant functor from $H$ to the category of sets, where the object $n$ of $H$ is mapped to the set of isotopy classes of embedding of the genus $n$ handlebody into $M$, compatible with the embeddings of the disc into the boundaries. This functor can be regarded as a refinement of the fundamental group of $M$, and we call it the quantum fundamental group of $M$. Using this invariant, we can construct for each co-ribbon Hopf algebra $A$ an invariant of 3-manifolds which may be regarded as (the space of regular functions on) the representation variety of $M$ with respect to $A$.

11月2日 -- 056号室, 16:30 -- 18:00

Daniel Ruberman (Brandeis University)

Periodic-end manifolds and SW theory.

Abstract: We study an extension of Seiberg-Witten invariants to 4-manifolds with the homology of S^1 \times S^3. This extension has many potential applications in low-dimensional topology, including the study of the homology cobordism group. Because b_2^+ =0, the usual strategy for defining invariants does not work--one cannot disregard reducible solutions. In fact, the count of solutions can jump in a family of metrics or perturbations. To remedy this, we define an index-theoretic counter-term that jumps by the same amount. The counterterm is the index of the Dirac operator on a manifold with a periodic end modeled at infinity by the infinite cyclic cover of the manifold. This is joint work with Tomasz Mrowka and Nikolai Saveliev.

11月9日 -- 056号室, 17:00 -- 18:00

大鹿健一 (大阪大学)

Characterising bumping points on deformation spaces of Kleinian groups

Abstract: Klein群の変形空間はその内部の相異なる成分がbump，あるいは同一成分が bumpすることがあることが知られている． Anderson-Canary-McCulloughの研究により，いかなる成分がbumpするかはわかっている．本講演ではどのような点でbumpするのかの条件を与える．

11月16日 -- 056号室, 16:30 -- 18:00

伊藤昇 (早稲田大学)

On a colored Khovanov bicomplex

Abstract: Jones 多項式の Khovanov ホモロジーと関連理論が近年活発に研究されている．Jones 多項式の一般化である colored Jones多項式についても Khovanov により対応するコホモロジーが導入され，特に Mackaay と Turner や Beliakova と Wehrli の研究を通し発展した．しかし，このコホモロジーが持つ２つの境界作用素によって，Khovanov 型の複体で２重複体となるものが構成できるのかは問題として残されていた．もしあるならば Khovanov 型のホモロジーが Total complex のコホモロジーに収束するスペクトル系列の第２項として理解される．この問題意識は Beliakova と Wehli の論文によって紹介された．今回はそれに対して一つの答えを与える．また似た文脈で colored Jones 多項式の別のスペクトル系列からは絡み目の colored Rasmussen 不変量が自然に出てくることも時間が許せば紹介したい．

11月30日 -- 056号室, 16:30 -- 18:00

中村信裕 (東京大学大学院数理科学研究科)

Pin^-(2)-monopole equations and intersection forms with local coefficients of 4-manifolds

Abstract: We introduce a variant of the Seiberg-Witten equations, Pin^-(2)-monopole equations, and explain its applications to intersection forms with local coefficients of 4-manifolds. The first application is an analogue of Froyshov's results on 4-manifolds which have definite forms with local coefficients. The second one is a local coefficient version of Furuta's 10/8-inequality. As a corollary, we construct nonsmoothable spin 4-manifolds satisfying Rohlin's theorem and the 10/8-inequality.

12月7日 -- 056号室, 16:30 -- 18:00

Raphael Ponge (東京大学大学院数理科学研究科)

Diffeomorphism-invariant geometries and noncommutative geometry.

Abstract: In many geometric situations we may encounter the action of a group $G$ on a manifold $M$, e.g., in the context of foliations. If the action is free and proper, then the quotient $M/G$ is a smooth manifold. However, in general the quotient $M/G$ need not even be Hausdorff. Furthermore, it is well-known that a manifold has structure invariant under the full group of diffeomorphisms except the differentiable structure itself. Under these conditions how can one do diffeomorphism-invariant geometry?
Noncommutative geometry provides us with the solution of trading the ill-behaved space $M/G$ for a non-commutative algebra which essentially plays the role of the algebra of smooth functions on that space. The local index formula of Atiyah-Singer ultimately holds in the setting of noncommutative geometry. Using this framework Connes and Moscovici then obtained in the 90s a striking reformulation of the local index formula in diffeomorphism-invariant geometry.
An important part the talk will be devoted to reviewing noncommutative geometry and Connes-Moscovici's index formula. We will then hint to on- going projects about reformulating the local index formula in two new geometric settings: biholomorphism-invariant geometry of strictly pseudo-convex domains and contactomorphism-invariant geometry.

12月14日 -- 056号室, 16:30 -- 18:00

Kenneth Schackleton (IPMU)

On the coarse geometry of Teichmueller space

Abstract: We discuss the synthetic geometry of the pants graph in comparison with the Weil-Petersson metric, whose geometry the pants graph coarsely models following work of Brock’s. We also restrict our attention to the 5-holed sphere, studying the Gromov bordification of the pants graph and the dynamics of pseudo-Anosov mapping classes.

1月11日 -- 056号室, 16:30 -- 18:00

河澄響矢 (東京大学大学院数理科学研究科)

The Chas-Sullivan conjecture for a surface of infinite genus

Abstract: 久野雄介氏（広島大理、学振PD）との共同研究。 \Sigma_{\infty,1} を境界成分 1 の向きづけられたコンパクト曲面の帰納極限とする。この曲面 \Sigma_{\infty,1} の Goldman Lie 代数の中心が定数ループで張られることを証明する。閉曲面についての同様の定理を Chas と Sullivan が予想し、Etingof が証明している。我々の結果は向きづけられたコード図式の Lie 代数の中心を計算することで証明される。時間が許せば、線型コード図式の空間上の Lie 代数の構造についても議論したい。

1月25日 -- 056号室, 16:30 -- 17:30

春田力 (東京大学大学院数理科学研究科)

シート数が小さい曲面結び目の自明化について

Abstract: A connected surface smoothly embedded in ${\mathbb R}^4$ is called a surface-knot. In particular, if a surface-knot $F$ is homeomorphic to the $2$-sphere or the torus, then it is called an $S^2$-knot or a $T^2$-knot, respectively. The sheet number of a surface-knot is an invariant analogous to the crossing number of a $1$-knot. M. Saito and S. Satoh proved some results concerning the sheet number of an $S^2$-knot. In particular, it is known that an $S^2$-knot is trivial if and only if its sheet number is $1$, and there is no $S^2$-knot whose sheet number is $2$. In this talk, we show that there is no $S^2$-knot whose sheet number is $3$, and a $T^2$-knot is trivial if and only if its sheet number is $1$.