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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
2016年11月29日(火)
17:30-18:30 数理科学研究科棟(駒場) 056号室
Tea: Common Room 17:00-17:30
千葉 逸人 氏 (九州大学)
一般化スペクトル理論とその結合振動子系のダイナミクスへの応用 (JAPANESE)
Tea: Common Room 17:00-17:30
千葉 逸人 氏 (九州大学)
一般化スペクトル理論とその結合振動子系のダイナミクスへの応用 (JAPANESE)
[ 講演概要 ]
一般のグラフの上で定義された大自由度結合振動子系のダイナミクスを考える。特に、結合強度を大きくしていくと非同期状態から同期状態への相転移が起こることを、一般化スペクトル理論を用いて示す。
一般のグラフの上で定義された大自由度結合振動子系のダイナミクスを考える。特に、結合強度を大きくしていくと非同期状態から同期状態への相転移が起こることを、一般化スペクトル理論を用いて示す。
2016年11月22日(火)
17:00-18:30 数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
内藤 貴仁 氏 (東京大学大学院数理科学研究科)
Sullivan's coproduct on the reduced loop homology (JAPANESE)
Tea: Common Room 16:30-17:00
内藤 貴仁 氏 (東京大学大学院数理科学研究科)
Sullivan's coproduct on the reduced loop homology (JAPANESE)
[ 講演概要 ]
In string topology, Sullivan introduced a coproduct on the reduced loop homology and showed that the homology has an infinitesimal bialgebra structure with respect to the coproduct and Chas-Sullivan loop product. In this talk, I will give a homotopy theoretic description of Sullivan's coproduct. By using the description, we obtain some computational examples of the structure over the rational number field. Moreover, I will also discuss a based loop space version of the coproduct.
In string topology, Sullivan introduced a coproduct on the reduced loop homology and showed that the homology has an infinitesimal bialgebra structure with respect to the coproduct and Chas-Sullivan loop product. In this talk, I will give a homotopy theoretic description of Sullivan's coproduct. By using the description, we obtain some computational examples of the structure over the rational number field. Moreover, I will also discuss a based loop space version of the coproduct.
2016年11月15日(火)
17:00-18:30 数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
逆井 卓也 氏 (東京大学大学院数理科学研究科)
Cohomology of the moduli space of graphs and groups of homology cobordisms of surfaces (JAPANESE)
Tea: Common Room 16:30-17:00
逆井 卓也 氏 (東京大学大学院数理科学研究科)
Cohomology of the moduli space of graphs and groups of homology cobordisms of surfaces (JAPANESE)
[ 講演概要 ]
We construct an abelian quotient of the symplectic derivation Lie algebra of the free Lie algebra generated by the fundamental representation of the symplectic group. It gives an alternative proof of the fact first shown by Bartholdi that the top rational homology group of the moduli space of metric graphs of rank 7 is one dimensional. As an application, we construct a non-trivial abelian quotient of the homology cobordism group of a surface of positive genus. This talk is based on joint works with Shigeyuki Morita, Masaaki Suzuki and Gwénaël Massuyeau.
We construct an abelian quotient of the symplectic derivation Lie algebra of the free Lie algebra generated by the fundamental representation of the symplectic group. It gives an alternative proof of the fact first shown by Bartholdi that the top rational homology group of the moduli space of metric graphs of rank 7 is one dimensional. As an application, we construct a non-trivial abelian quotient of the homology cobordism group of a surface of positive genus. This talk is based on joint works with Shigeyuki Morita, Masaaki Suzuki and Gwénaël Massuyeau.
2016年11月08日(火)
17:00-18:30 数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
秋田 利之 氏 (北海道大学)
Second mod 2 homology of Artin groups (JAPANESE)
Tea: Common Room 16:30-17:00
秋田 利之 氏 (北海道大学)
Second mod 2 homology of Artin groups (JAPANESE)
[ 講演概要 ]
After a brief survey on the K($\pi$,1) conjecture and homology of Artin groups,I will introduce our recent result: we determined second mod 2 homology of arbitrary Artin groups without assuming the K($\pi$,1)-conjecture. The key ingredients are Hopf's formula and a result of Howlett on Schur multipliers of Coxeter groups. This is a joint work with Ye Liu.
After a brief survey on the K($\pi$,1) conjecture and homology of Artin groups,I will introduce our recent result: we determined second mod 2 homology of arbitrary Artin groups without assuming the K($\pi$,1)-conjecture. The key ingredients are Hopf's formula and a result of Howlett on Schur multipliers of Coxeter groups. This is a joint work with Ye Liu.
2016年11月01日(火)
17:00-18:30 数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
大場 貴裕 氏 (東京工業大学)
Higher-dimensional contact manifolds with infinitely many Stein fillings (JAPANESE)
Tea: Common Room 16:30-17:00
大場 貴裕 氏 (東京工業大学)
Higher-dimensional contact manifolds with infinitely many Stein fillings (JAPANESE)
[ 講演概要 ]
A Stein fillings of a given contact manifold is a Stein domain whose boundary is contactomorphic to the given contact manifold.
Open books, Lefschetz fibrations, and mapping class groups of their fibers in particular help us to produce various contact manifolds and their Stein fillings. However, little is known about mapping class groups of higher-dimensional manifolds. This is one of the reasons that it was unknown whether there is a contact manifold of dimension > 3 with infinitely many Stein fillings. In this talk, I will choose a certain symplectic manifold as fibers of open books and Lefschetz fibrations and by using them, construct an infinite family of higher-dimensional contact manifolds with infinitely many Stein fillings.
A Stein fillings of a given contact manifold is a Stein domain whose boundary is contactomorphic to the given contact manifold.
Open books, Lefschetz fibrations, and mapping class groups of their fibers in particular help us to produce various contact manifolds and their Stein fillings. However, little is known about mapping class groups of higher-dimensional manifolds. This is one of the reasons that it was unknown whether there is a contact manifold of dimension > 3 with infinitely many Stein fillings. In this talk, I will choose a certain symplectic manifold as fibers of open books and Lefschetz fibrations and by using them, construct an infinite family of higher-dimensional contact manifolds with infinitely many Stein fillings.
2016年10月18日(火)
17:30-18:30 数理科学研究科棟(駒場) 056号室
Tea: Common Room 17:00-17:30
橋本 義武 氏 (東京都市大学)
拡大W代数に対する共形場理論 (JAPANESE)
Tea: Common Room 17:00-17:30
橋本 義武 氏 (東京都市大学)
拡大W代数に対する共形場理論 (JAPANESE)
[ 講演概要 ]
This talk is based on a joint work with A. Tsuchiya (Kavli IPMU) and T. Matsumoto (Nagoya Univ). In 2006 Feigin-Gainutdinov-Semikhatov-Tipunin introduced vertex operator algebras M called extended W-algebras. Tsuchiya-Wood developed representation theory of M by the method of
"infinitesimal deformation of parameter" and Jack symmetric polynomials.
In this talk I will discuss the following subjects:
1. "factorization" in conformal field theory,
2. tensor structure of the category of M-modules and "module-bimodule correspondence".
This talk is based on a joint work with A. Tsuchiya (Kavli IPMU) and T. Matsumoto (Nagoya Univ). In 2006 Feigin-Gainutdinov-Semikhatov-Tipunin introduced vertex operator algebras M called extended W-algebras. Tsuchiya-Wood developed representation theory of M by the method of
"infinitesimal deformation of parameter" and Jack symmetric polynomials.
In this talk I will discuss the following subjects:
1. "factorization" in conformal field theory,
2. tensor structure of the category of M-modules and "module-bimodule correspondence".
2016年10月11日(火)
17:00-18:30 数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
河澄 響矢 氏 (東京大学大学院数理科学研究科)
The Kashiwara-Vergne problem and the Goldman-Turaev Lie bialgebra in genus zero (JAPANESE)
Tea: Common Room 16:30-17:00
河澄 響矢 氏 (東京大学大学院数理科学研究科)
The Kashiwara-Vergne problem and the Goldman-Turaev Lie bialgebra in genus zero (JAPANESE)
[ 講演概要 ]
In view of results of Goldman and Turaev, the free vector space over the free loops on an oriented surface has a natural Lie bialgebra structure. The Goldman bracket has a formal description by using a special (or symplectic) expansion of the fundamental group of the surface. It is natural to ask for a formal description of the Turaev cobracket. We will show how to obtain a formal description of the Goldman-Turaev Lie bialgebra for genus 0 using a solution of the Kashiwara-Vergne problem. A similar description was recently obtained by Massuyeau using the Kontsevich integral. Moreover we propose a generalization of the Kashiwara-Vergne problem in the context of the Goldman-Turaev Lie bialgebra. This talk is based on a joint work with A. Alekseev, Y. Kuno and F. Naef.
In view of results of Goldman and Turaev, the free vector space over the free loops on an oriented surface has a natural Lie bialgebra structure. The Goldman bracket has a formal description by using a special (or symplectic) expansion of the fundamental group of the surface. It is natural to ask for a formal description of the Turaev cobracket. We will show how to obtain a formal description of the Goldman-Turaev Lie bialgebra for genus 0 using a solution of the Kashiwara-Vergne problem. A similar description was recently obtained by Massuyeau using the Kontsevich integral. Moreover we propose a generalization of the Kashiwara-Vergne problem in the context of the Goldman-Turaev Lie bialgebra. This talk is based on a joint work with A. Alekseev, Y. Kuno and F. Naef.
2016年09月27日(火)
17:00-18:30 数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
藤内 翔太 氏 (東京大学大学院数理科学研究科)
CAT(0) properties for orthoscheme complexes (JAPANESE)
Tea: Common Room 16:30-17:00
藤内 翔太 氏 (東京大学大学院数理科学研究科)
CAT(0) properties for orthoscheme complexes (JAPANESE)
[ 講演概要 ]
Gromov showed that a cubical complex is locally CAT(0) if and only if the link of every vertex is a flag complex. Brady and MacCammond introduced an orthoscheme complex as a generalization of cubical complexes. It is, however, difficult to tell whether an orthoscheme complex is (locally) CAT(0) or not. In this talk, I will discuss a translation of Gromov's characterization for orthoscheme complexes. As a generalization of Gromov's characterization, I will show that the orthoscheme complex of locally distributive semilattice is CAT(0) if and only if it is a flag semilattice.
Gromov showed that a cubical complex is locally CAT(0) if and only if the link of every vertex is a flag complex. Brady and MacCammond introduced an orthoscheme complex as a generalization of cubical complexes. It is, however, difficult to tell whether an orthoscheme complex is (locally) CAT(0) or not. In this talk, I will discuss a translation of Gromov's characterization for orthoscheme complexes. As a generalization of Gromov's characterization, I will show that the orthoscheme complex of locally distributive semilattice is CAT(0) if and only if it is a flag semilattice.
2016年07月19日(火)
17:00-18:30 数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
渡邊 陽介 氏 (University of Hawaii)
The geometry of the curve graphs and beyond (JAPANESE)
Tea: Common Room 16:30-17:00
渡邊 陽介 氏 (University of Hawaii)
The geometry of the curve graphs and beyond (JAPANESE)
[ 講演概要 ]
The curve graphs are locally infinite. However, by using Masur-Minsky's tight geodesics, one could view them as locally finite graphs. Bell-Fujiwara used a special property of tight geodesics and showed that the asymptotic dimension of the curve graphs is finite. In this talk, I will introduce a new class of geodesics which also has the property. If time permits, I will explain how such geodesics can be adapted in Out(F_n) setting.
The curve graphs are locally infinite. However, by using Masur-Minsky's tight geodesics, one could view them as locally finite graphs. Bell-Fujiwara used a special property of tight geodesics and showed that the asymptotic dimension of the curve graphs is finite. In this talk, I will introduce a new class of geodesics which also has the property. If time permits, I will explain how such geodesics can be adapted in Out(F_n) setting.
2016年07月12日(火)
17:00-18:30 数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
John Parker 氏 (Durham University)
Non-arithmetic lattices (ENGLISH)
Tea: Common Room 16:30-17:00
John Parker 氏 (Durham University)
Non-arithmetic lattices (ENGLISH)
[ 講演概要 ]
In this talk I will discuss arithmetic and non-arithmetic lattices and I will give a history of the problem of finding non-arithmetic lattices. I will also briefly describe the construction of new non-arithmetic lattices in SU(2,1) found in my joint workwith Martin Deraux and Julien Paupert.
In this talk I will discuss arithmetic and non-arithmetic lattices and I will give a history of the problem of finding non-arithmetic lattices. I will also briefly describe the construction of new non-arithmetic lattices in SU(2,1) found in my joint workwith Martin Deraux and Julien Paupert.
2016年06月28日(火)
17:00-18:30 数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
見村 万佐人 氏 (東北大学)
Strong algebraization of fixed point properties (JAPANESE)
Tea: Common Room 16:30-17:00
見村 万佐人 氏 (東北大学)
Strong algebraization of fixed point properties (JAPANESE)
[ 講演概要 ]
バナッハ空間（ないしは族）を固定したとき，有限生成群のそれ上の等長作用が常に大域的固定点を持つ，という性質を固定点性質と呼ぶ．ヒルベルト空間全体のなす族を考えたときの固定点性質は，「Kazhdan の性質(T)」と呼ばれる群の剛性と同値であることが知られている．
離散群の線型表現の分類は連続群と違い，群が少しでも複雑になると手に負えない．これが原因で，離散群の固定点性質を直接示すことは当面の間著しく困難であった．Y. Shalom は1999年の論文（Publ. IHES）で，固定点性質を部分群に分けて，最後に“パッチワーク”する，という手法を応用し，上の困難に対し初のブレイクスルーをもたらした．しかし，Shalomのパッチワーク戦略では群の部分群による「有界生成（Bounded Generation）」という厄介な要請が本質的であって（後述するように実はこれは気のせいだったのだが，長年そう信じられてきたように講演者には思われる），この要請がShalomの手法を適用する際の致命的な弱点となっていた．
今回，講演者はShalomのパッチワーク（1999,2006）の思想を発展させて，「有界生成」条件を舞台から追いやることに成功した．講演者の条件は，
部分群たちを広げていくある“ゲーム”の必勝戦略として記述される．講演ではこの“ゲーム”の内容・証明のあらすじをお話したい．これにより，
「有界生成」の成立がわからないような状況でもパッチワーク戦略を適用できうるようになった．系として，いろいろな離散群が強い固定点性質をもつことを示せ，しかも証明も非常にコンセプチュアルである．こうした応用面についても概観したい．
バナッハ空間（ないしは族）を固定したとき，有限生成群のそれ上の等長作用が常に大域的固定点を持つ，という性質を固定点性質と呼ぶ．ヒルベルト空間全体のなす族を考えたときの固定点性質は，「Kazhdan の性質(T)」と呼ばれる群の剛性と同値であることが知られている．
離散群の線型表現の分類は連続群と違い，群が少しでも複雑になると手に負えない．これが原因で，離散群の固定点性質を直接示すことは当面の間著しく困難であった．Y. Shalom は1999年の論文（Publ. IHES）で，固定点性質を部分群に分けて，最後に“パッチワーク”する，という手法を応用し，上の困難に対し初のブレイクスルーをもたらした．しかし，Shalomのパッチワーク戦略では群の部分群による「有界生成（Bounded Generation）」という厄介な要請が本質的であって（後述するように実はこれは気のせいだったのだが，長年そう信じられてきたように講演者には思われる），この要請がShalomの手法を適用する際の致命的な弱点となっていた．
今回，講演者はShalomのパッチワーク（1999,2006）の思想を発展させて，「有界生成」条件を舞台から追いやることに成功した．講演者の条件は，
部分群たちを広げていくある“ゲーム”の必勝戦略として記述される．講演ではこの“ゲーム”の内容・証明のあらすじをお話したい．これにより，
「有界生成」の成立がわからないような状況でもパッチワーク戦略を適用できうるようになった．系として，いろいろな離散群が強い固定点性質をもつことを示せ，しかも証明も非常にコンセプチュアルである．こうした応用面についても概観したい．
2016年06月21日(火)
17:00-18:30 数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
伊藤 昇 氏 (東京大学大学院数理科学研究科)
Spaces of chord diagrams of spherical curves (JAPANESE)
Tea: Common Room 16:30-17:00
伊藤 昇 氏 (東京大学大学院数理科学研究科)
Spaces of chord diagrams of spherical curves (JAPANESE)
[ 講演概要 ]
In this talk, the speaker introduces a framework to obtain (possibly infinitely many) new topological invariants of spherical curves under local homotopy moves (several types of Reidemeister moves). They are defined by chord diagrams, each of which is a configurations of even paired points on a circle. We see that these invariants have useful properties.
In this talk, the speaker introduces a framework to obtain (possibly infinitely many) new topological invariants of spherical curves under local homotopy moves (several types of Reidemeister moves). They are defined by chord diagrams, each of which is a configurations of even paired points on a circle. We see that these invariants have useful properties.
2016年06月14日(火)
17:00-18:30 数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
粕谷 直彦 氏 (青山学院大学)
Non-Kähler complex structures on R^4 (JAPANESE)
Tea: Common Room 16:30-17:00
粕谷 直彦 氏 (青山学院大学)
Non-Kähler complex structures on R^4 (JAPANESE)
[ 講演概要 ]
We consider the following problem. "Is there any non-Kähler complex structure on R^{2n}?" If n=1, the answer is clearly negative. On the other hand, Calabi and Eckmann constructed non-Kähler complex structures on R^{2n} for n ≥ 3. In this talk, I will construct uncountably many non-Kähler complex structures on R^4, and give the affirmative answer to the case where n=2. For the construction, it is important to understand the genus-one achiral Lefschetz fibration S^4 → S^2 found by Yukio Matsumoto and Kenji Fukaya. This is a joint work with Antonio Jose Di Scala and Daniele Zuddas.
We consider the following problem. "Is there any non-Kähler complex structure on R^{2n}?" If n=1, the answer is clearly negative. On the other hand, Calabi and Eckmann constructed non-Kähler complex structures on R^{2n} for n ≥ 3. In this talk, I will construct uncountably many non-Kähler complex structures on R^4, and give the affirmative answer to the case where n=2. For the construction, it is important to understand the genus-one achiral Lefschetz fibration S^4 → S^2 found by Yukio Matsumoto and Kenji Fukaya. This is a joint work with Antonio Jose Di Scala and Daniele Zuddas.
2016年06月07日(火)
17:00-18:30 数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
早野 健太 氏 (慶應義塾大学)
Topology of holomorphic Lefschetz pencils on the four-torus (JAPANESE)
Tea: Common Room 16:30-17:00
早野 健太 氏 (慶應義塾大学)
Topology of holomorphic Lefschetz pencils on the four-torus (JAPANESE)
[ 講演概要 ]
In this talk, we will show that two holomorphic Lefschetz pencils on the four-torus are (smoothly) isomorphic if they have the same genus and divisibility. The proof relies on the theory of moduli spaces of polarized abelian surfaces. We will also give vanishing cycles of some holomorphic pencils of the four-torus explicitly. This is joint work with Noriyuki Hamada (The University of Tokyo).
In this talk, we will show that two holomorphic Lefschetz pencils on the four-torus are (smoothly) isomorphic if they have the same genus and divisibility. The proof relies on the theory of moduli spaces of polarized abelian surfaces. We will also give vanishing cycles of some holomorphic pencils of the four-torus explicitly. This is joint work with Noriyuki Hamada (The University of Tokyo).
2016年05月31日(火)
17:00-18:30 数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
Benoît Guerville-Ballé 氏 (東京学芸大学)
A linking invariant for algebraic curves (ENGLISH)
Tea: Common Room 16:30-17:00
Benoît Guerville-Ballé 氏 (東京学芸大学)
A linking invariant for algebraic curves (ENGLISH)
[ 講演概要 ]
We construct a topological invariant of algebraic plane curves, which is in some sense an adaptation of the linking number of knot theory. As an application, we show that this invariant distinguishes a new Zariski pair of curves (ie a pair of curves having same combinatorics, yet different topology).
We construct a topological invariant of algebraic plane curves, which is in some sense an adaptation of the linking number of knot theory. As an application, we show that this invariant distinguishes a new Zariski pair of curves (ie a pair of curves having same combinatorics, yet different topology).
2016年05月24日(火)
17:00-18:30 数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
田中 心 氏 (東京学芸大学)
Independence of Roseman moves for surface-knot diagrams (JAPANESE)
Tea: Common Room 16:30-17:00
田中 心 氏 (東京学芸大学)
Independence of Roseman moves for surface-knot diagrams (JAPANESE)
[ 講演概要 ]
Roseman moves are seven types of local modifications for surface-knot diagrams in 3-space which generate ambient isotopies of surface-knots in 4-space. In this talk, I will discuss independence among the seven Roseman moves. In particular, I will focus on Roseman moves involving triple points and on those involving branch points. The former is joint work with Kanako Oshiro (Sophia University) and Kengo Kawamura (Osaka City University), and the latter is joint work with Masamichi Takase (Seikei University).
Roseman moves are seven types of local modifications for surface-knot diagrams in 3-space which generate ambient isotopies of surface-knots in 4-space. In this talk, I will discuss independence among the seven Roseman moves. In particular, I will focus on Roseman moves involving triple points and on those involving branch points. The former is joint work with Kanako Oshiro (Sophia University) and Kengo Kawamura (Osaka City University), and the latter is joint work with Masamichi Takase (Seikei University).
2016年05月17日(火)
17:00-18:30 数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
正井 秀俊 氏 (東京大学大学院数理科学研究科)
Some dynamics of random walks on the mapping class groups (JAPANESE)
Tea: Common Room 16:30-17:00
正井 秀俊 氏 (東京大学大学院数理科学研究科)
Some dynamics of random walks on the mapping class groups (JAPANESE)
[ 講演概要 ]
The dynamics of random walks on the mapping class groups on closed surfaces of genus >1 will be discussed. We define the topological entropy of random walks. Then we prove that the drift with respect to Thurston or Teichmüller metrics and the Lyapunov exponent all coincide with the topological entropy. This is a "random version" of pseudo-Anosov dynamics observed by Thurston and I will begin this talk by recalling the work of Thurston.
The dynamics of random walks on the mapping class groups on closed surfaces of genus >1 will be discussed. We define the topological entropy of random walks. Then we prove that the drift with respect to Thurston or Teichmüller metrics and the Lyapunov exponent all coincide with the topological entropy. This is a "random version" of pseudo-Anosov dynamics observed by Thurston and I will begin this talk by recalling the work of Thurston.
2016年05月10日(火)
17:00-18:30 数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
小鳥居 祐香 氏 (東京大学大学院数理科学研究科)
On Milnor's link-homotopy invariants for handlebody-links (JAPANESE)
Tea: Common Room 16:30-17:00
小鳥居 祐香 氏 (東京大学大学院数理科学研究科)
On Milnor's link-homotopy invariants for handlebody-links (JAPANESE)
[ 講演概要 ]
A handlebody-link is a disjoint union of handlebodies embedded in $S^3$ and HL-homotopy is an equivalence relation on handlebody-links generated by self-crossing changes. A. Mizusawa and R. Nikkuni classified the set of HL-homotopy classes of 2-component handlebody-links completely using the linking numbers for handlebody-links. In this talk, by using Milnor's link-homotopy invariants, we construct an invariant for handlebody-links and give a bijection between the set of HL-homotopy classes of n-component handlebody-links with some assumption and a quotient of the action of the general linear group on a tensor product of modules. This is joint work with Atsuhiko Mizusawa at Waseda University.
A handlebody-link is a disjoint union of handlebodies embedded in $S^3$ and HL-homotopy is an equivalence relation on handlebody-links generated by self-crossing changes. A. Mizusawa and R. Nikkuni classified the set of HL-homotopy classes of 2-component handlebody-links completely using the linking numbers for handlebody-links. In this talk, by using Milnor's link-homotopy invariants, we construct an invariant for handlebody-links and give a bijection between the set of HL-homotopy classes of n-component handlebody-links with some assumption and a quotient of the action of the general linear group on a tensor product of modules. This is joint work with Atsuhiko Mizusawa at Waseda University.
2016年04月26日(火)
17:00-18:30 数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
植木 潤 氏 (東京大学大学院数理科学研究科)
Arithmetic topology on branched covers of 3-manifolds (JAPANESE)
Tea: Common Room 16:30-17:00
植木 潤 氏 (東京大学大学院数理科学研究科)
Arithmetic topology on branched covers of 3-manifolds (JAPANESE)
[ 講演概要 ]
The analogy between 3-dimensional topology and number theory was first pointed out by Mazur in the 1960s, and it has been studied systematically by Kapranov, Reznikov, Morishita, and others. In their analogies, for example, knots and 3-manifolds correspond to primes and number rings respectively. The study of these analogies is called arithmetic topology now.
In my talk, based on their dictionary of analogies, we study analogues of idelic class field theory, Iwasawa theory, and Galois deformation theory in the context of 3-dimensional topology, and establish various foundational analogies in arithmetic topology.
The analogy between 3-dimensional topology and number theory was first pointed out by Mazur in the 1960s, and it has been studied systematically by Kapranov, Reznikov, Morishita, and others. In their analogies, for example, knots and 3-manifolds correspond to primes and number rings respectively. The study of these analogies is called arithmetic topology now.
In my talk, based on their dictionary of analogies, we study analogues of idelic class field theory, Iwasawa theory, and Galois deformation theory in the context of 3-dimensional topology, and establish various foundational analogies in arithmetic topology.
2016年04月19日(火)
17:00-18:30 数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
Błażej Szepietowski 氏 (Gdansk University)
Topological rigidity of finite cyclic group actions on compact surfaces (ENGLISH)
Tea: Common Room 16:30-17:00
Błażej Szepietowski 氏 (Gdansk University)
Topological rigidity of finite cyclic group actions on compact surfaces (ENGLISH)
[ 講演概要 ]
Two actions of a group on a surface are called topologically equivalent if they are conjugate by a homeomorphism of the surface. I will describe a method of enumeration (and classification) of topological equivalence classes of actions of a finite group on a compact surface, based on the combinatorial theory of noneuclidean crystallographic groups (NEC groups in short) and a relationship between the outer automorphism group of an NEC group and certain mapping class group. By this method we study topological equivalence of actions of a finite cyclic group on a compact surface, in the situation where the order of the group is large relative to the genus of the surface.
Two actions of a group on a surface are called topologically equivalent if they are conjugate by a homeomorphism of the surface. I will describe a method of enumeration (and classification) of topological equivalence classes of actions of a finite group on a compact surface, based on the combinatorial theory of noneuclidean crystallographic groups (NEC groups in short) and a relationship between the outer automorphism group of an NEC group and certain mapping class group. By this method we study topological equivalence of actions of a finite cyclic group on a compact surface, in the situation where the order of the group is large relative to the genus of the surface.
2016年04月12日(火)
17:00-18:30 数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30 - 17:00
Aniceto Murillo 氏 (Universidad de Malaga)
Homotopy theory of differential graded Lie algebras (ENGLISH)
Tea: Common Room 16:30 - 17:00
Aniceto Murillo 氏 (Universidad de Malaga)
Homotopy theory of differential graded Lie algebras (ENGLISH)
[ 講演概要 ]
Having as motivation the Deligne's principle by which every deformation functor is governed by a differential graded Lie algebra, we build a homotopy theory for these algebras which extend the classical Quillen approach and let us model any (non necessarily 1-connected nor path connected) complex. This is joint work with Urtzi Buijs, Yves Félix and Daniel Tanré.
Having as motivation the Deligne's principle by which every deformation functor is governed by a differential graded Lie algebra, we build a homotopy theory for these algebras which extend the classical Quillen approach and let us model any (non necessarily 1-connected nor path connected) complex. This is joint work with Urtzi Buijs, Yves Félix and Daniel Tanré.
2016年04月05日(火)
17:00-18:30 数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
北山 貴裕 氏 (東京大学大学院数理科学研究科)
Torsion invariants and representation varieties for non-positively curved cube complexes (JAPANESE)
Tea: Common Room 16:30-17:00
北山 貴裕 氏 (東京大学大学院数理科学研究科)
Torsion invariants and representation varieties for non-positively curved cube complexes (JAPANESE)
[ 講演概要 ]
Applications of torsion invariants and representation varieties have been extensively studied for 3-manifolds. Twisted Alexander polynomials are known to detect the Thurston norm and fiberedness of a 3-manifold. Ideal points of character varieties are known to detect essential surfaces in a 3-manifold in a certain extension of Culler-Shalen theory. In view of cubulation of 3-manifolds one can expect that these results naturally extend to a wider framework and, in particular, the case of virtually special cube complexes. We formulate and discuss such analogous questions for non-positively curved cube complexes.
Applications of torsion invariants and representation varieties have been extensively studied for 3-manifolds. Twisted Alexander polynomials are known to detect the Thurston norm and fiberedness of a 3-manifold. Ideal points of character varieties are known to detect essential surfaces in a 3-manifold in a certain extension of Culler-Shalen theory. In view of cubulation of 3-manifolds one can expect that these results naturally extend to a wider framework and, in particular, the case of virtually special cube complexes. We formulate and discuss such analogous questions for non-positively curved cube complexes.
2016年02月16日(火)
17:00-18:30 数理科学研究科棟(駒場) 056号室
Tea : Common Room 16:30 -- 17:00
Luc Menichi 氏 (University of Angers)
String Topology, Euler Class and TNCZ free loop fibrations (ENGLISH)
Tea : Common Room 16:30 -- 17:00
Luc Menichi 氏 (University of Angers)
String Topology, Euler Class and TNCZ free loop fibrations (ENGLISH)
[ 講演概要 ]
Let $M$ be a connected, closed oriented manifold.
Chas and Sullivan have defined a loop product and a loop coproduct on
$H_*(LM;¥mathbb{F})$, the homology of the
free loops on $M$ with coefficients in the field $¥mathbb{F}$.
By studying this loop coproduct, I will show that if the free loop
fibration
$¥Omega M¥buildrel{i}¥over¥hookrightarrow
LM¥buildrel{ev}¥over¥twoheadrightarrow M$
is homologically trivial, i.e. $i^*:H^*(LM;¥mathbb{F})¥twoheadrightarrow
H^*(¥Omega M;¥mathbb{F})$ is onto,
then the Euler characteristic of $M$ is divisible by the characteristic
of the field $¥mathbb{F}$
(or $M$ is a point).
Let $M$ be a connected, closed oriented manifold.
Chas and Sullivan have defined a loop product and a loop coproduct on
$H_*(LM;¥mathbb{F})$, the homology of the
free loops on $M$ with coefficients in the field $¥mathbb{F}$.
By studying this loop coproduct, I will show that if the free loop
fibration
$¥Omega M¥buildrel{i}¥over¥hookrightarrow
LM¥buildrel{ev}¥over¥twoheadrightarrow M$
is homologically trivial, i.e. $i^*:H^*(LM;¥mathbb{F})¥twoheadrightarrow
H^*(¥Omega M;¥mathbb{F})$ is onto,
then the Euler characteristic of $M$ is divisible by the characteristic
of the field $¥mathbb{F}$
(or $M$ is a point).
2016年01月19日(火)
15:00-16:00 数理科学研究科棟(駒場) 056号室
山本 光 氏 (東京大学大学院数理科学研究科)
Ricci-mean curvature flows in gradient shrinking Ricci solitons (JAPANESE)
山本 光 氏 (東京大学大学院数理科学研究科)
Ricci-mean curvature flows in gradient shrinking Ricci solitons (JAPANESE)
[ 講演概要 ]
A Ricci-mean curvature flow is a coupled parabolic PDE system of a mean
curvature flow and a Ricci flow.
In this talk, we consider a Ricci-mean curvature flow in a gradient
shrinking Ricci soliton, and give a generalization of a well-known result
of Huisken which states that if a mean curvature flow in a Euclidean space
develops a singularity of type I, then its parabolic rescaling near the singular
point converges to a self-shrinker.
A Ricci-mean curvature flow is a coupled parabolic PDE system of a mean
curvature flow and a Ricci flow.
In this talk, we consider a Ricci-mean curvature flow in a gradient
shrinking Ricci soliton, and give a generalization of a well-known result
of Huisken which states that if a mean curvature flow in a Euclidean space
develops a singularity of type I, then its parabolic rescaling near the singular
point converges to a self-shrinker.
2016年01月12日(火)
16:30-18:30 数理科学研究科棟(駒場) 056号室
川崎 盛通 氏 (東京大学大学院数理科学研究科) 16:30-17:30
重い部分集合と非可縮周期軌道 (JAPANESE)
On codimension two contact embeddings in the standard spheres (JAPANESE)
川崎 盛通 氏 (東京大学大学院数理科学研究科) 16:30-17:30
重い部分集合と非可縮周期軌道 (JAPANESE)
[ 講演概要 ]
ビランとポルテロヴィッチ、サラモンによる研究では、開シンプレクティック多
様体Mとその部分集合$X$, $M$内の自由ホモトピー類αに対する相対的なシンプレクテ
ィック容量$C_{BPS}(M,X,α)$を定義した。
$C_{BPS}(M,X,α)$はM上のハミルトン函数がXで十分大きい値を取る場合にαを代表
する周期軌道が存在するかという問題に関わって定義される。
一方で、エントフとポルテロヴィッチは非交叉配置性の文脈でシンプレクティッ
ク多様体の「重い」部分集合というものを定義している。
本講演ではビラン・ポルテロヴィッチ・サラモン容量$C_{BPS}(M,X,α)$の有限性
(適当な設定下での周期軌道の存在)を重い部分集合を用いて示す方法について解
説する。
これまでの研究では(自由ループの)ホモトピー類αを代表する周期軌道の検出に
は、αを代表する軌道のハミルトン・フレアー理論を用いるのが一般的であった。
重い部分集合は(可縮軌道のハミルトン・フレアー理論の)スペクトル不変量を用
いて定義される概念であるので、今回の手法では可縮軌道のハミルトン・フレア
ー理論を用いて非可縮軌道を検出することになる。
古川 遼 氏 (東京大学大学院数理科学研究科) 17:30-18:30ビランとポルテロヴィッチ、サラモンによる研究では、開シンプレクティック多
様体Mとその部分集合$X$, $M$内の自由ホモトピー類αに対する相対的なシンプレクテ
ィック容量$C_{BPS}(M,X,α)$を定義した。
$C_{BPS}(M,X,α)$はM上のハミルトン函数がXで十分大きい値を取る場合にαを代表
する周期軌道が存在するかという問題に関わって定義される。
一方で、エントフとポルテロヴィッチは非交叉配置性の文脈でシンプレクティッ
ク多様体の「重い」部分集合というものを定義している。
本講演ではビラン・ポルテロヴィッチ・サラモン容量$C_{BPS}(M,X,α)$の有限性
(適当な設定下での周期軌道の存在)を重い部分集合を用いて示す方法について解
説する。
これまでの研究では(自由ループの)ホモトピー類αを代表する周期軌道の検出に
は、αを代表する軌道のハミルトン・フレアー理論を用いるのが一般的であった。
重い部分集合は(可縮軌道のハミルトン・フレアー理論の)スペクトル不変量を用
いて定義される概念であるので、今回の手法では可縮軌道のハミルトン・フレア
ー理論を用いて非可縮軌道を検出することになる。
On codimension two contact embeddings in the standard spheres (JAPANESE)
[ 講演概要 ]
In this talk we consider codimension two contact
embedding problem by using higher dimensional braids.
First, we focus on embeddings of contact $3$-manifolds to the standard $
S^5$ and give some results, for example, any contact structure on $S^3$
can embed so that it is smoothly isotopic to the standard embedding.
These are joint work with John Etnyre. Second, we consider the relative
Euler number of codimension two contact submanifolds and its Seifert
hypersurfaces which is a generalization of the self-linking number of
transverse knots in contact $3$-manifolds. We give a way to calculate
the relative Euler number of certain contact submanifolds obtained by
braids and as an application we give examples of embeddings of one
contact manifold which are isotopic as smooth embeddings but not
isotopic as contact embeddings in higher dimension.
In this talk we consider codimension two contact
embedding problem by using higher dimensional braids.
First, we focus on embeddings of contact $3$-manifolds to the standard $
S^5$ and give some results, for example, any contact structure on $S^3$
can embed so that it is smoothly isotopic to the standard embedding.
These are joint work with John Etnyre. Second, we consider the relative
Euler number of codimension two contact submanifolds and its Seifert
hypersurfaces which is a generalization of the self-linking number of
transverse knots in contact $3$-manifolds. We give a way to calculate
the relative Euler number of certain contact submanifolds obtained by
braids and as an application we give examples of embeddings of one
contact manifold which are isotopic as smooth embeddings but not
isotopic as contact embeddings in higher dimension.
