Topology

2013年10月 -- 2014年3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 8, 2014
世話係　
河野俊丈
河澄響矢

10月1日 -- 056号室, 16:30 -- 18:00

門田直之 (東京理科大学)

The geography problem of Lefschetz fibrations

Abstract: To consider holomorphic fibrations complex surfaces over complex curves and Lefschetz fibrations over surfaces is one method for the study of complex surfaces of general type and symplectic 4-manifods, respectively. In this talk, by comparing the geography problem of relatively minimal holomorphic fibrations with that of relatively minimal Lefschetz fibrations (i.e., the characterization of pairs (x,y) of certain invariants x and ycorresponding to relatively minimal holomorphic fibrations and relatively minimal Lefschetz fibrations), we observe the difference between complex surfaces of general type and symplectic 4-manifolds. In particular, we construct Lefschetz fibrations violating the "slope inequality" which holds for any relatively minimal holomorphic fibrations.

10月8日 -- 056号室, 16:30 -- 18:00

清水達郎 (東京大学大学院数理科学研究科)

An invariant of rational homology 3-spheres via vector fields.

Abstract: In this talk, we define an invariant of rational homology 3-spheres with values in a space $\mathcal A(\emptyset)$ of Jacobi diagrams by using vector fields. The construction of our invariant is a generalization of both that of the Kontsevich-Kuperberg-Thurston invariant $z^{KKT}$ and that of Fukaya and Watanabe's Morse homotopy invariant $z^{FW}$. As an application of our invariant, we prove that $z^{KKT}=z^{FW}$ for integral homology 3-spheres.

10月15日 -- 056号室, 16:30 -- 18:00

高瀬将道 (成蹊大学)

Desingularizing special generic maps

Abstract: This is a joint work with Osamu Saeki (IMI, Kyushu University). A special generic map is a generic map which has only definite fold as its singularities. We study the condition for a special generic map from a closed n-manifold to the p-space (n+1>p), to factor through a codimension one immersion (or an embedding). In particular, for the cases where p = 1 and 2 we obtain complete results. Our techniques are related to Smale-Hirsch theory, topology of the space of immersions, relation between the space of topological immersions and that of smooth immersions, sphere eversions, differentiable structures of homotopy spheres, diffeomorphism group of spheres, free group actions on the sphere, etc.

10月22日 -- 056号室, 16:30 -- 18:00

井上玲 (千葉大学)

Cluster algebra and complex volume of knots

Abstract: The cluster algebra was introduced by Fomin and Zelevinsky around 2000. The characteristic operation in the algebra called 'mutation' is related to various notions in mathematics and mathematical physics. In this talk I review a basics of the cluster algebra, and introduce its application to study the complex volume of knot complements in S³. Here a mutation corresponds to an ideal tetrahedron. This talk is based on joint work with Kazuhiro Hikami (Kyushu University).

10月29日 -- 056号室, 16:30 -- 18:00

Daniel Matei (IMAR, Bucharest)

Fundamental groups of algebraic varieties

Abstract: We discuss restrictions imposed by the complex structure on fundamental groups of quasi-projective algebraic varieties with mild singularities. We investigate quasi-projectivity of various geometric classes of finitely presented groups.

11月5日 -- 123号室, 16:30 -- 18:00

Carlos Moraga Ferrándiz (東京大学大学院数理科学研究科, 日本学術振興会)

The isotopy problem of non-singular closed 1-forms.

Abstract: Given alpha_0, alpha_1 two cohomologous non-singular closed 1-forms of a compact manifold M, are they always isotopic? We expect a negative answer to this question, at least in high dimensions by the work of Laudenbach, as well as an obstruction living in the algebraic K-theory of the Novikov ring associated to the underlying cohomology class. A similar problem for functions N x [0,1] --> [0,1] without critical points was treated by Hatcher and Wagoner in the 70s.
The first goal of this talk is to explain how we can carry a part of the strategy of Hatcher and Wagoner into the context of closed 1-forms and to indicate the main difficulties that appear by doing so. The second goal is to show the techniques to treat this difficulties and the progress in defining the expected obstruction.

11月12日 -- 056号室, 16:30 -- 18:00

Alexander Voronov (University of Minnesota)

The Batalin-Vilkovisky Formalism and Cohomology of Moduli Spaces

Abstract: We use the Batalin-Vilkovisky formalism to give a new proof of Costello's theorem on the existence and uniqueness of solution to the Quantum Master Equation. We also make a physically motivated conjecture on the rational homology of moduli spaces. This is a joint work with Domenico D'Alessandro.

11月19日 -- 056号室, 16:30 -- 18:00

児玉大樹 (東京大学大学院数理科学研究科)

測度論的基本領域を持つ円周上の極小微分同相写像

Abstract: 任意の無理数αに対して、ルベーグ測度について基本領域を持つ円周上の極小微分同相写像で回転数がαとなるものを構成した。
これは松元重則氏（日本大学）との共同研究である。

11月26日 -- 056号室, 16:30 -- 18:00

徳永浩雄 (首都大学東京)

有理楕円曲面とあるline-conic arrangements

Abstract: Sは有理楕円曲面とする．Sの生成ファイバーは 1変数有理函数体上の楕円曲線であり，楕円曲線の群構造を利用してSの切断C_1からS上の曲線 C_2を構成することできる．本講演では，このアイデアに基づいて得られるある7次のline-conic arrangementsについて解説する．

12月3日 -- 056号室, 17:00 -- 18:00

Bruno Martelli (Università di Pisa)

Hyperbolic four-manifolds with one cusp

Abstract: (joint work with A. Kolpakov)
We introduce a simple algorithm which transforms every four-dimensional cubulation into a cusped finite-volume hyperbolic four-manifold. Combinatorially distinct cubulations give rise to topologically distinct manifolds. Using this algorithm we construct the first examples of finite-volume hyperbolic four-manifolds with one cusp. More generally, we show that the number of k-cusped hyperbolic four-manifolds with volume smaller than V grows like C^{V log V} for any fixed k. As a corollary, we deduce that the 3-torus bounds geometrically a hyperbolic manifold.

Martelli氏の講演はキャンセルになりました。

12月10日 -- 056号室, 16:30 -- 18:00

丹下基生 (筑波大学)

Corks, plugs, and local moves of 4-manifolds.

Abstract: Akbulut and Yasui defined cork, and plug to produce many exotic pairs. In this talk, we introduce a plug with respect to Fintushel-Stern's knot surgery or more 4-dimensional local moves and and argue by using Heegaard Fleor theory.

12月17日 -- 056号室, 16:30 -- 18:00

中村伊南沙 (東京大学大学院数理科学研究科)

Satellites of an oriented surface link and their local moves

Abstract: For an oriented surface link F in R⁴, we consider a satellite construction of a surface link, called a 2-dimensional braid over F, which is in the form of a covering over F. We introduce the notion of an m-chart on a surface diagram p(F) &sub R³ of F, which is a finite graph on p(F) satisfying certain conditions and is an extended notion of an m-chart on a 2-disk presenting a surface braid. A 2-dimensional braid over F is presented by an m-chart on p(F). It is known that two surface links are equivalent if and only if their surface diagrams are related by a finite sequence of ambient isotopies of R³ and local moves called Roseman moves. We show that Roseman moves for surface diagrams with m-charts can be well-defined. Further, we give some applications.

12月24日 -- 056号室, 16:30 -- 18:00

Tirasan Khandhawit (Kavli IPMU)

Stable homotopy type for monopole Floer homology

Abstract: In this talk, I will try to give an overview of the construction of stable homotopy type for monopole Floer homology. The construction associates a stable homotopy object to 3-manifolds, which will recover the Floer groups by appropriate homology theory. The main ingredients are finite dimensional approximation technique and Conley index theory. In addition, I will demonstrate construction for certain 3-manifolds such as the 3-torus.

1月14日 -- 056号室, 17:00 -- 18:00

Rinat Kashaev (University of Geneva)

State-integral partition functions on shaped triangulations

Abstract: Quantum Teichmüller theory can be promoted to a generalized TQFT within the combinatorial framework of shaped triangulations with the tetrahedral weight functions given in terms of the Weil-Gelfand-Zak transformation of Faddeev.FN"s quantum dilogarithm. By using simple examples, I will illustrate the connection of this theory with the hyperbolic geometry in three dimensions.

1月21日 -- 056号室

16:30 -- 17:30

粕谷直彦 (東京大学大学院数理科学研究科)

On contact submanifolds of the odd dimensional Euclidean space

Abstract: We prove that the Chern class of a closed contact manifold is an obstruction for codimension two contact embeddings in the odd dimensional Euclidean space. By Gromov's h-principle, for any closed contact 3-manifold with trivial first Chern class, there is a contact structure on R⁵ which admits a contact embedding.

17:30 -- 18:30

李暁龍 (東京大学大学院数理科学研究科)

ホモクリニック類における弱固有値：小さい角度を持つサドルからの摂動

Abstract: For 3-dimensional homoclinic classes of saddles with index 2, a new sufficient condition for creating weak contracting eigenvalues is provided. Our perturbation makes use of small angles between stable and unstable subspaces of saddles. In particular, by recovering the unstable eigenvector, we can designate that the newly created weak eigenvalue is contracting. As applications, we obtain C¹-generic non-trivial index- intervals of homoclinic classes and the C¹-approximation of robust heterodimensional cycles. In particular, this sufficient condition is satisfied by a substantial class of saddles with homoclinic tangencies.