本文印刷
全画面プリント
トポロジー火曜セミナー
過去の記録 ～04/19｜次回の予定｜今後の予定 04/20～
| 開催情報 | 火曜日 17:00～18:30 数理科学研究科棟(駒場) 056号室 |
|---|---|
| 担当者 | 河澄 響矢, 北山 貴裕, 逆井卓也 |
| セミナーURL | http://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index.html |
過去の記録
2014年01月21日(火)
16:30-17:30 数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
粕谷 直彦 氏 (東京大学大学院数理科学研究科)
On contact submanifolds of the odd dimensional Euclidean space (JAPANESE)
Tea: 16:00 - 16:30 コモンルーム
粕谷 直彦 氏 (東京大学大学院数理科学研究科)
On contact submanifolds of the odd dimensional Euclidean space (JAPANESE)
[ 講演概要 ]
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 $\\mathbb{R}^5$ which admits a contact
embedding.
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 $\\mathbb{R}^5$ which admits a contact
embedding.
2014年01月21日(火)
17:30-18:30 数理科学研究科棟(駒場) 056号室
李 暁龍 氏 (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 (ENGLISH)
李 暁龍 氏 (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 (ENGLISH)
[ 講演概要 ]
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 C^1-generic non-trivial index-
intervals of homoclinic classes and the C^1-approximation of robust
heterodimensional cycles. In particular, this sufficient condition is
satisfied by a substantial class of saddles with homoclinic tangencies.
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 C^1-generic non-trivial index-
intervals of homoclinic classes and the C^1-approximation of robust
heterodimensional cycles. In particular, this sufficient condition is
satisfied by a substantial class of saddles with homoclinic tangencies.
2014年01月14日(火)
17:00-18:00 数理科学研究科棟(駒場) 056号室
Tea: 16:30 - 17:00 コモンルーム
Rinat Kashaev 氏 (University of Geneva)
State-integral partition functions on shaped triangulations (ENGLISH)
Tea: 16:30 - 17:00 コモンルーム
Rinat Kashaev 氏 (University of Geneva)
State-integral partition functions on shaped triangulations (ENGLISH)
[ 講演概要 ]
Quantum Teichm\\"uller 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.
Quantum Teichm\\"uller 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.
2013年12月24日(火)
16:30-18:00 数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
Tirasan Khandhawit 氏 (Kavli IPMU)
Stable homotopy type for monopole Floer homology (ENGLISH)
Tea: 16:00 - 16:30 コモンルーム
Tirasan Khandhawit 氏 (Kavli IPMU)
Stable homotopy type for monopole Floer homology (ENGLISH)
[ 講演概要 ]
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.
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.
2013年12月17日(火)
16:30-18:00 数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
中村 伊南沙 氏 (東京大学大学院数理科学研究科)
Satellites of an oriented surface link and their local moves (JAPANESE)
Tea: 16:00 - 16:30 コモンルーム
中村 伊南沙 氏 (東京大学大学院数理科学研究科)
Satellites of an oriented surface link and their local moves (JAPANESE)
[ 講演概要 ]
For an oriented surface link $F$ in $\\mathbb{R}^4$,
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
$p(F)\\subset \\mathbb{R}^3$ 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 $\\mathbb{R}^3$ 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.
For an oriented surface link $F$ in $\\mathbb{R}^4$,
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
$p(F)\\subset \\mathbb{R}^3$ 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 $\\mathbb{R}^3$ 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.
2013年12月10日(火)
16:30-18:00 数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
丹下 基生 氏 (筑波大学)
Corks, plugs, and local moves of 4-manifolds. (JAPANESE)
Tea: 16:00 - 16:30 コモンルーム
丹下 基生 氏 (筑波大学)
Corks, plugs, and local moves of 4-manifolds. (JAPANESE)
[ 講演概要 ]
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.
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.
2013年12月03日(火)
17:00-18:00 数理科学研究科棟(駒場) 056号室
Tea: 16:30 - 17:00 コモンルーム
Bruno Martelli 氏 (Univ. di Pisa)
Hyperbolic four-manifolds with one cusp (cancelled) (JAPANESE)
Tea: 16:30 - 17:00 コモンルーム
Bruno Martelli 氏 (Univ. di Pisa)
Hyperbolic four-manifolds with one cusp (cancelled) (JAPANESE)
[ 講演概要 ]
(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.
(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.
2013年11月26日(火)
16:30-18:00 数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
徳永 浩雄 氏 (首都大学東京)
有理楕円曲面とあるline-conic arrangements (JAPANESE)
Tea: 16:00 - 16:30 コモンルーム
徳永 浩雄 氏 (首都大学東京)
有理楕円曲面とあるline-conic arrangements (JAPANESE)
[ 講演概要 ]
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.
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.
2013年11月19日(火)
16:30-18:00 数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
児玉 大樹 氏 (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 (JAPANESE)
Tea: 16:00 - 16:30 コモンルーム
児玉 大樹 氏 (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 (JAPANESE)
[ 講演概要 ]
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。
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。
2013年11月12日(火)
16:30-18:00 数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
Alexander Voronov 氏 (University of Minnesota)
The Batalin-Vilkovisky Formalism and Cohomology of Moduli Spaces (ENGLISH)
Tea: 16:00 - 16:30 コモンルーム
Alexander Voronov 氏 (University of Minnesota)
The Batalin-Vilkovisky Formalism and Cohomology of Moduli Spaces (ENGLISH)
[ 講演概要 ]
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.
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.
2013年11月05日(火)
16:30-18:00 数理科学研究科棟(駒場) 123号室
Tea: 16:00 - 16:30 コモンルーム
Carlos Moraga Ferrandiz 氏 (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. (ENGLISH)
Tea: 16:00 - 16:30 コモンルーム
Carlos Moraga Ferrandiz 氏 (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. (ENGLISH)
[ 講演概要 ]
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.
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.
2013年10月29日(火)
16:30-18:00 数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
Daniel Matei 氏 (IMAR, Bucharest)
Fundamental groups of algebraic varieties (ENGLISH)
Tea: 16:00 - 16:30 コモンルーム
Daniel Matei 氏 (IMAR, Bucharest)
Fundamental groups of algebraic varieties (ENGLISH)
[ 講演概要 ]
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.
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.
2013年10月22日(火)
16:30-18:00 数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
井上 玲 氏 (千葉大学)
Cluster algebra and complex volume of knots (JAPANESE)
Tea: 16:00 - 16:30 コモンルーム
井上 玲 氏 (千葉大学)
Cluster algebra and complex volume of knots (JAPANESE)
[ 講演概要 ]
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 S^3.
Here a mutation corresponds to an ideal tetrahedron.
This talk is based on joint work with Kazuhiro Hikami (Kyushu University).
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 S^3.
Here a mutation corresponds to an ideal tetrahedron.
This talk is based on joint work with Kazuhiro Hikami (Kyushu University).
2013年10月15日(火)
16:30-18:00 数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
高瀬 将道 氏 (成蹊大学)
Desingularizing special generic maps (JAPANESE)
Tea: 16:00 - 16:30 コモンルーム
高瀬 将道 氏 (成蹊大学)
Desingularizing special generic maps (JAPANESE)
[ 講演概要 ]
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.
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.
2013年10月08日(火)
16:30-18:00 数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
清水 達郎 氏 (東京大学大学院数理科学研究科 )
An invariant of rational homology 3-spheres via vector fields. (JAPANESE)
Tea: 16:00 - 16:30 コモンルーム
清水 達郎 氏 (東京大学大学院数理科学研究科 )
An invariant of rational homology 3-spheres via vector fields. (JAPANESE)
[ 講演概要 ]
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.
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.
2013年10月01日(火)
16:30-18:00 数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
門田 直之 氏 (東京理科大学)
The geography problem of Lefschetz fibrations (JAPANESE)
Tea: 16:00 - 16:30 コモンルーム
門田 直之 氏 (東京理科大学)
The geography problem of Lefschetz fibrations (JAPANESE)
[ 講演概要 ]
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 $y$ corresponding 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.
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 $y$ corresponding 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.
2013年07月16日(火)
17:10-18:10 数理科学研究科棟(駒場) 056号室
Tea: 16:50 - 17:10 コモンルーム
山田 澄生 氏 (学習院大学)
実双曲空間の新しいモデルについて (JAPANESE)
Tea: 16:50 - 17:10 コモンルーム
山田 澄生 氏 (学習院大学)
実双曲空間の新しいモデルについて (JAPANESE)
[ 講演概要 ]
本講演ではクラインおよびポアンカレ以来位相幾何学の発展に伴って多くの重要な空間を提供してきた実双曲空間の実現について、 道具立ては古典的ではあるが新しいモデルを紹介する。それらの構成法は凸幾何学と射影幾何学と密接に関連しており、数学史の観点 からも興味深いと思われる。これはAthanase Papadopoulosとの共同研究である。
本講演ではクラインおよびポアンカレ以来位相幾何学の発展に伴って多くの重要な空間を提供してきた実双曲空間の実現について、 道具立ては古典的ではあるが新しいモデルを紹介する。それらの構成法は凸幾何学と射影幾何学と密接に関連しており、数学史の観点 からも興味深いと思われる。これはAthanase Papadopoulosとの共同研究である。
2013年07月09日(火)
16:30-18:00 数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
Ryan Budney 氏 (University of Victoria)
Smooth 3-manifolds in the 4-sphere (ENGLISH)
Tea: 16:00 - 16:30 コモンルーム
Ryan Budney 氏 (University of Victoria)
Smooth 3-manifolds in the 4-sphere (ENGLISH)
[ 講演概要 ]
Everyone who has studied topology knows the compact 2-manifolds that embed in the 3-sphere. One dimension up, the problem of which smooth 3-manifolds embed in the 4-sphere turns out to be much more involved with a handful of partial answers. I will describe what is known at the present moment.
Everyone who has studied topology knows the compact 2-manifolds that embed in the 3-sphere. One dimension up, the problem of which smooth 3-manifolds embed in the 4-sphere turns out to be much more involved with a handful of partial answers. I will describe what is known at the present moment.
2013年06月25日(火)
17:10-18:10 数理科学研究科棟(駒場) 056号室
Tea: 16:50 - 17:10 コモンルーム
渡邉 忠之 氏 (島根大学)
Higher-order generalization of Fukaya's Morse homotopy
invariant of 3-manifolds (JAPANESE)
Tea: 16:50 - 17:10 コモンルーム
渡邉 忠之 氏 (島根大学)
Higher-order generalization of Fukaya's Morse homotopy
invariant of 3-manifolds (JAPANESE)
[ 講演概要 ]
In his article published in 1996, K. Fukaya constructed
a 3-manifold invariant by using Morse homotopy theory. Roughly, his
invariant is defined by considering several Morse functions on a
3-manifold and counting with weights the ways that the theta-graph can
be immersed such that edges follow gradient lines. We generalize his
construction to 3-valent graphs with arbitrary number of loops for
integral homology 3-spheres. I will also discuss extension of our method
to 3-manifolds with positive first Betti numbers.
In his article published in 1996, K. Fukaya constructed
a 3-manifold invariant by using Morse homotopy theory. Roughly, his
invariant is defined by considering several Morse functions on a
3-manifold and counting with weights the ways that the theta-graph can
be immersed such that edges follow gradient lines. We generalize his
construction to 3-valent graphs with arbitrary number of loops for
integral homology 3-spheres. I will also discuss extension of our method
to 3-manifolds with positive first Betti numbers.
2013年06月18日(火)
16:30-18:00 数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
茂手木 公彦 氏 (日本大学)
Left-orderable, non-L-space surgeries on knots (JAPANESE)
Tea: 16:00 - 16:30 コモンルーム
茂手木 公彦 氏 (日本大学)
Left-orderable, non-L-space surgeries on knots (JAPANESE)
[ 講演概要 ]
A Dehn surgery is said to be left-orderable
if the resulting manifold of the surgery has the left-orderable fundamental group,
and a Dehn surgery is called an L-space surgery
if the resulting manifold of the surgery is an L-space.
We will focus on left-orderable, non-L-space surgeries on knots in the 3-sphere.
Once we have a knot with left-orderable surgeries,
the ``periodic construction" enables us to provide infinitely many knots with
left-orderable, non-L-space surgeries.
We apply the construction to present infinitely many hyperbolic knots on each
of which every nontrivial surgery is a left-orderable, non-L-space surgery.
This is a joint work with Masakazu Teragaito.
A Dehn surgery is said to be left-orderable
if the resulting manifold of the surgery has the left-orderable fundamental group,
and a Dehn surgery is called an L-space surgery
if the resulting manifold of the surgery is an L-space.
We will focus on left-orderable, non-L-space surgeries on knots in the 3-sphere.
Once we have a knot with left-orderable surgeries,
the ``periodic construction" enables us to provide infinitely many knots with
left-orderable, non-L-space surgeries.
We apply the construction to present infinitely many hyperbolic knots on each
of which every nontrivial surgery is a left-orderable, non-L-space surgery.
This is a joint work with Masakazu Teragaito.
2013年06月11日(火)
16:30-18:00 数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
北山 貴裕 氏 (東京大学大学院数理科学研究科)
On an analogue of Culler-Shalen theory for higher-dimensional
representations
(JAPANESE)
Tea: 16:00 - 16:30 コモンルーム
北山 貴裕 氏 (東京大学大学院数理科学研究科)
On an analogue of Culler-Shalen theory for higher-dimensional
representations
(JAPANESE)
[ 講演概要 ]
Culler and Shalen established a way to construct incompressible surfaces
in a 3-manifold from ideal points of the SL_2-character variety. We
present an analogous theory to construct certain kinds of branched
surfaces from limit points of the SL_n-character variety. Such a
branched surface induces a nontrivial presentation of the fundamental
group as a 2-dimensional complex of groups. This is a joint work with
Takashi Hara (Osaka University).
Culler and Shalen established a way to construct incompressible surfaces
in a 3-manifold from ideal points of the SL_2-character variety. We
present an analogous theory to construct certain kinds of branched
surfaces from limit points of the SL_n-character variety. Such a
branched surface induces a nontrivial presentation of the fundamental
group as a 2-dimensional complex of groups. This is a joint work with
Takashi Hara (Osaka University).
2013年06月04日(火)
16:30-18:00 数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
Mustafa Korkmaz 氏 (Middle East Technical University)
Low-dimensional linear representations of mapping class groups. (ENGLISH)
Tea: 16:00 - 16:30 コモンルーム
Mustafa Korkmaz 氏 (Middle East Technical University)
Low-dimensional linear representations of mapping class groups. (ENGLISH)
[ 講演概要 ]
For a compact connected orientable surface, the mapping class group
of it is defined as the group of isotopy classes of orientation-preserving
self-diffeomorphisms of S which are identity on the boundary. The action
of the mapping class group on the first homology of the surface
gives rise to the classical 2g-dimensional symplectic representation.
The existence of a faithful linear representation of the mapping class
group is still unknown. In my talk, I will show the following three results;
there is no lower dimensional (complex) linear representation,
up to conjugation the symplectic representation is the unique nontrivial representation in dimension 2g, and there is no faithful linear representation
of the mapping class group in dimensions up to 3g-3. I will also discuss a few applications of these theorems, including some algebraic consequences.
For a compact connected orientable surface, the mapping class group
of it is defined as the group of isotopy classes of orientation-preserving
self-diffeomorphisms of S which are identity on the boundary. The action
of the mapping class group on the first homology of the surface
gives rise to the classical 2g-dimensional symplectic representation.
The existence of a faithful linear representation of the mapping class
group is still unknown. In my talk, I will show the following three results;
there is no lower dimensional (complex) linear representation,
up to conjugation the symplectic representation is the unique nontrivial representation in dimension 2g, and there is no faithful linear representation
of the mapping class group in dimensions up to 3g-3. I will also discuss a few applications of these theorems, including some algebraic consequences.
2013年05月21日(火)
16:30-18:00 数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
Yuanyuan Bao 氏 (東京大学大学院数理科学研究科)
A Heegaard Floer homology for bipartite spatial graphs and its
properties (ENGLISH)
Tea: 16:00 - 16:30 コモンルーム
Yuanyuan Bao 氏 (東京大学大学院数理科学研究科)
A Heegaard Floer homology for bipartite spatial graphs and its
properties (ENGLISH)
[ 講演概要 ]
A spatial graph is a smooth embedding of a graph into a given
3-manifold. We can regard a link as a particular spatial graph.
So it is natural to ask whether it is possible to extend the idea
of link Floer homology to define a Heegaard Floer homology for
spatial graphs. In this talk, we discuss some ideas towards this
question. In particular, we define a Heegaard Floer homology for
bipartite spatial graphs and discuss some further observations
about this construction. We remark that Harvey and O’Donnol
have announced a combinatorial Floer homology for spatial graphs by
considering grid diagrams.
A spatial graph is a smooth embedding of a graph into a given
3-manifold. We can regard a link as a particular spatial graph.
So it is natural to ask whether it is possible to extend the idea
of link Floer homology to define a Heegaard Floer homology for
spatial graphs. In this talk, we discuss some ideas towards this
question. In particular, we define a Heegaard Floer homology for
bipartite spatial graphs and discuss some further observations
about this construction. We remark that Harvey and O’Donnol
have announced a combinatorial Floer homology for spatial graphs by
considering grid diagrams.
2013年05月14日(火)
16:30-18:00 数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
早野 健太 氏 (大阪大学)
Vanishing cycles and homotopies of wrinkled fibrations (JAPANESE)
Tea: 16:00 - 16:30 コモンルーム
早野 健太 氏 (大阪大学)
Vanishing cycles and homotopies of wrinkled fibrations (JAPANESE)
[ 講演概要 ]
Wrinkled fibrations on closed 4-manifolds are stable
maps to closed surfaces with only indefinite singularities. Such
fibrations and variants of them have been studied for the past few years
to obtain new descriptions of 4-manifolds using mapping class groups.
Vanishing cycles of wrinkled fibrations play a key role in these studies.
In this talk, we will explain how homotopies of wrinkled fibrtions affect
their vanishing cycles. Part of the results in this talk is a joint work
with Stefan Behrens (Max Planck Institute for Mathematics).
Wrinkled fibrations on closed 4-manifolds are stable
maps to closed surfaces with only indefinite singularities. Such
fibrations and variants of them have been studied for the past few years
to obtain new descriptions of 4-manifolds using mapping class groups.
Vanishing cycles of wrinkled fibrations play a key role in these studies.
In this talk, we will explain how homotopies of wrinkled fibrtions affect
their vanishing cycles. Part of the results in this talk is a joint work
with Stefan Behrens (Max Planck Institute for Mathematics).
2013年05月07日(火)
16:30-18:00 数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
伊藤 哲也 氏 (京都大学数理解析研究所)
Homological intersection in braid group representation and dual
Garside structure (JAPANESE)
Tea: 16:00 - 16:30 コモンルーム
伊藤 哲也 氏 (京都大学数理解析研究所)
Homological intersection in braid group representation and dual
Garside structure (JAPANESE)
[ 講演概要 ]
One method to construct linear representations of braid groups is to use
an action of braid groups on certain homology of local system coefficient.
Many famous representations, such as Burau or Lawrence-Krammer-Bigelow
representations are constructed in such a way. We show that homological
intersections on such homology groups are closely related to the dual
Garside structure, a remarkable combinatorial structure of braid, and
prove that some representations detects the length of braids in a
surprisingly simple way.
This work is partially joint with Bert Wiest (Univ. Rennes1).
One method to construct linear representations of braid groups is to use
an action of braid groups on certain homology of local system coefficient.
Many famous representations, such as Burau or Lawrence-Krammer-Bigelow
representations are constructed in such a way. We show that homological
intersections on such homology groups are closely related to the dual
Garside structure, a remarkable combinatorial structure of braid, and
prove that some representations detects the length of braids in a
surprisingly simple way.
This work is partially joint with Bert Wiest (Univ. Rennes1).
