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トポロジー火曜セミナー
過去の記録 ～07/26｜次回の予定｜今後の予定 07/27～
| 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
2018年10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
Daniel Matei 氏 (IMAR Bucharest)
Resonance varieties and matrix tree theorems (ENGLISH)
Tea: Common Room 16:30-17:00
Daniel Matei 氏 (IMAR Bucharest)
Resonance varieties and matrix tree theorems (ENGLISH)
[ 講演概要 ]
We discuss the resonance varieties, encoding vanishing of cohomology cup products, of various classes of finitely presented groups of geometric and combinatorial origin. We describe the ideals defining those varieties in terms spanning trees in a similar vein with the classical matrix tree theorem in graph theory. We present applications of this description to 3-manifold groups and Artin groups.
We discuss the resonance varieties, encoding vanishing of cohomology cup products, of various classes of finitely presented groups of geometric and combinatorial origin. We describe the ideals defining those varieties in terms spanning trees in a similar vein with the classical matrix tree theorem in graph theory. We present applications of this description to 3-manifold groups and Artin groups.
2018年10月09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
Boris Hasselblatt 氏 (Tufts University)
Foulon surgery, new contact flows, and dynamical complexity (ENGLISH)
Tea: Common Room 16:30-17:00
Boris Hasselblatt 氏 (Tufts University)
Foulon surgery, new contact flows, and dynamical complexity (ENGLISH)
[ 講演概要 ]
A refinement of Dehn surgery produces new contact flows that are unusual and interesting in several ways. The geodesic flow of a hyperbolic surface becomes a nonalgebraic contact Anosov flow with larger orbit growth, and the purely periodic fiber flow becomes parabolic or hyperbolic. Moreover, Reeb flows for other contact forms for the same contact structure have the same complexity. Finally, an idea by Vinhage promises a quantification of the complexity increase.
A refinement of Dehn surgery produces new contact flows that are unusual and interesting in several ways. The geodesic flow of a hyperbolic surface becomes a nonalgebraic contact Anosov flow with larger orbit growth, and the purely periodic fiber flow becomes parabolic or hyperbolic. Moreover, Reeb flows for other contact forms for the same contact structure have the same complexity. Finally, an idea by Vinhage promises a quantification of the complexity increase.
2018年10月02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
鮑 園園 氏 (東京大学大学院数理科学研究科)
An Alexander polynomial for MOY graphs (JAPANESE)
Tea: Common Room 16:30-17:00
鮑 園園 氏 (東京大学大学院数理科学研究科)
An Alexander polynomial for MOY graphs (JAPANESE)
[ 講演概要 ]
An MOY graph is a trivalent graph equipped with a balanced coloring. In this talk, we define a version of Alexander polynomial for an MOY graph. This polynomial is the Euler characteristic of the Heegaard Floer homology of an MOY graph. We give a characterization of the polynomial, which we call MOY-type relations, and show that it is equivalent to Viro’s gl(1 | 1)-Alexander polynomial of a graph. (A part of the talk is a joint work of Zhongtao Wu)
An MOY graph is a trivalent graph equipped with a balanced coloring. In this talk, we define a version of Alexander polynomial for an MOY graph. This polynomial is the Euler characteristic of the Heegaard Floer homology of an MOY graph. We give a characterization of the polynomial, which we call MOY-type relations, and show that it is equivalent to Viro’s gl(1 | 1)-Alexander polynomial of a graph. (A part of the talk is a joint work of Zhongtao Wu)
2018年07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
石川 昌治 氏 (慶應義塾大学)
Positive flow-spines and contact 3-manifolds (JAPANESE)
Tea: Common Room 16:30-17:00
石川 昌治 氏 (慶應義塾大学)
Positive flow-spines and contact 3-manifolds (JAPANESE)
[ 講演概要 ]
A contact structure is a smooth distribution of hyperplanes on an odd-dimensional manifold that is non-integrable everywhere. In the case of dimension 3, there is a nice relationship between open book decompositions of 3-manifolds and contact structures up to contactomorphisms, called Giroux correspondence. A flow-spine is a spine of a 3-manifold admitting a flow such that it is transverse to the spine and the flow in the complement of the spine is diffeomorphic to a constant flow in an open ball. In this talk, we introduce some results in progress that give a correspondence between contact structures and positive flow-spines by regarding Reeb vector fields as flows of spines. This is a joint work with Y. Koda (Hiroshima) and H. Naoe (Tohoku).
A contact structure is a smooth distribution of hyperplanes on an odd-dimensional manifold that is non-integrable everywhere. In the case of dimension 3, there is a nice relationship between open book decompositions of 3-manifolds and contact structures up to contactomorphisms, called Giroux correspondence. A flow-spine is a spine of a 3-manifold admitting a flow such that it is transverse to the spine and the flow in the complement of the spine is diffeomorphic to a constant flow in an open ball. In this talk, we introduce some results in progress that give a correspondence between contact structures and positive flow-spines by regarding Reeb vector fields as flows of spines. This is a joint work with Y. Koda (Hiroshima) and H. Naoe (Tohoku).
2018年07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
Emmy Murphy 氏 (Northwestern University)
Loose Legendrians and arboreal singularities (ENGLISH)
Tea: Common Room 16:30-17:00
Emmy Murphy 氏 (Northwestern University)
Loose Legendrians and arboreal singularities (ENGLISH)
[ 講演概要 ]
Given a Stein manifold X, under what conditions can we ensure that X is symplectomorphic to C^n? For n>2 the condition of X being diffeomorphic to C^n does not suffice, and many counterexamples have been constructed which are detected by symplectic cohomology and the Fukaya category. One might conjecture that the diffeomorphism type together with a vanishing Fukaya category characterizes C^n. While this question is currently well of of reach, we present some new partial results. The main tools we'll discuss are arboreal singularities, constructable sheaf theory, and loose Legendrians -- and how they fit together to approach this question.
Given a Stein manifold X, under what conditions can we ensure that X is symplectomorphic to C^n? For n>2 the condition of X being diffeomorphic to C^n does not suffice, and many counterexamples have been constructed which are detected by symplectic cohomology and the Fukaya category. One might conjecture that the diffeomorphism type together with a vanishing Fukaya category characterizes C^n. While this question is currently well of of reach, we present some new partial results. The main tools we'll discuss are arboreal singularities, constructable sheaf theory, and loose Legendrians -- and how they fit together to approach this question.
2018年07月03日(火)
17:00-18:00 数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
吉田 純 氏 (東京大学大学院数理科学研究科)
Symmetries on algebras and Hochschild homology in view of categories of operators (JAPANESE)
Tea: Common Room 16:30-17:00
吉田 純 氏 (東京大学大学院数理科学研究科)
Symmetries on algebras and Hochschild homology in view of categories of operators (JAPANESE)
[ 講演概要 ]
The categorical construction of Hochschild homology by Connes reveals that the symmetric structure on the tensor product of abelian groups is essential. It means that the categorical meaning of ad-hoc generalizations of Hochschild homology in less symmetric monoidal abelian categories remains unclear. In this talk, I will propose formulation of this problem in terms of group operads introduced by Zhang. Moreover, for each group operad G, G-symmetric versions of categories of operators will be discussed. The notion plays a key role in defining Hochschild homology for homotopy algebras; such as topological Hochschild homology.
The categorical construction of Hochschild homology by Connes reveals that the symmetric structure on the tensor product of abelian groups is essential. It means that the categorical meaning of ad-hoc generalizations of Hochschild homology in less symmetric monoidal abelian categories remains unclear. In this talk, I will propose formulation of this problem in terms of group operads introduced by Zhang. Moreover, for each group operad G, G-symmetric versions of categories of operators will be discussed. The notion plays a key role in defining Hochschild homology for homotopy algebras; such as topological Hochschild homology.
2018年06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
今野 北斗 氏 (東京大学大学院数理科学研究科)
Characteristic classes via 4-dimensional gauge theory (JAPANESE)
Tea: Common Room 16:30-17:00
今野 北斗 氏 (東京大学大学院数理科学研究科)
Characteristic classes via 4-dimensional gauge theory (JAPANESE)
[ 講演概要 ]
Using gauge theory, more precisely SO(3)-Yang-Mills theory and Seiberg-Witten theory, I will construct characteristic classes of 4-manifold bundles. These characteristic classes are extensions of the SO(3)-Donaldson invariant and the Seiberg-Witten invariant to families of 4-manifolds, and can detect non-triviality of smooth 4-manifold bundles. The basic idea of the construction of these characteristic classes is to consider an infinite dimensional analogue of classical characteristic classes of manifold bundles, typified by the Mumford-Morita-Miller classes for surface bundles.
Using gauge theory, more precisely SO(3)-Yang-Mills theory and Seiberg-Witten theory, I will construct characteristic classes of 4-manifold bundles. These characteristic classes are extensions of the SO(3)-Donaldson invariant and the Seiberg-Witten invariant to families of 4-manifolds, and can detect non-triviality of smooth 4-manifold bundles. The basic idea of the construction of these characteristic classes is to consider an infinite dimensional analogue of classical characteristic classes of manifold bundles, typified by the Mumford-Morita-Miller classes for surface bundles.
2018年06月19日(火)
14:30-16:00 数理科学研究科棟(駒場) 056号室
RIKEN iTHEMS と共催
深谷 賢治 氏 (サイモンズセンター, SUNY)
相対かつ同変ラグランジアンフレアーホモロジーとアティヤ-フレアー予想 (JAPANESE)
RIKEN iTHEMS と共催
深谷 賢治 氏 (サイモンズセンター, SUNY)
相対かつ同変ラグランジアンフレアーホモロジーとアティヤ-フレアー予想 (JAPANESE)
[ 講演概要 ]
アティヤ-フレアー予想は，ゲージ理論におけるフレアーホモロジーとラグランジアンフレアーホモロジーの間の関係に関するものである．その一つの困難は，ラグランジアンフレアーホモロジーを考えるシンプレクティック多様体が特異点を持つことである．双対かつ同変ラグランジアンフレアーホモロジーを考えることで，この困難が解消し，少なくともアティヤ-フレアー予想を数学的に厳密な予想として定式化できることを説明する．
アティヤ-フレアー予想は，ゲージ理論におけるフレアーホモロジーとラグランジアンフレアーホモロジーの間の関係に関するものである．その一つの困難は，ラグランジアンフレアーホモロジーを考えるシンプレクティック多様体が特異点を持つことである．双対かつ同変ラグランジアンフレアーホモロジーを考えることで，この困難が解消し，少なくともアティヤ-フレアー予想を数学的に厳密な予想として定式化できることを説明する．
2018年06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
三松 佳彦 氏 (中央大学)
Turbulization of 2-dimensional foliations on 4-manifolds (JAPANESE)
Tea: Common Room 16:30-17:00
三松 佳彦 氏 (中央大学)
Turbulization of 2-dimensional foliations on 4-manifolds (JAPANESE)
[ 講演概要 ]
This is a report on a joint work with Elmar VOGT(Freie Universität Berlin). For codimension 1 foliations, the process of turbulization, i.e., inserting a Reeb component along a closed transversal, is well-known, while for higher codimensional foliation, similar processes were not understood until around 2006.
In this talk, first we formulate the turbulization along a closed transversal. Then in our dimension setting, namely 2-dimensional foliations on 4-manifolds ((4,2)-foliations), a cohomological criterion is given for a given transversal to a foliation, which tells the turbulization is possible or not, relying on the Thurston's h-principle. Also we give cocrete geometric constructions of turbulizations.
The cohomological criterion for turbulization is deduced from a more general criterion for a given embedded surface to be a compact leaf or a closed transversal of some foliation, which is stated in terms of the euler classes of tangent and normal bndle of the foliation to look for. The anormalous cohomological solutions for certain cases suggested the geometric realization of turbulization, while the cohomological criterion is obtained by the h-principle.
Some other modifications are also formulated for (4,2)-foliations and their possibility are assured by the anormalous solutions mentioned above. For some of them, good geometric realizations are not yet known. So far the difficulty lies on the problem of the connected components of the space of representations of the surface groups to Diff S^1.
If the time permits, some special features on the h-principle for 2-dimensional foliations are also explained.
This is a report on a joint work with Elmar VOGT(Freie Universität Berlin). For codimension 1 foliations, the process of turbulization, i.e., inserting a Reeb component along a closed transversal, is well-known, while for higher codimensional foliation, similar processes were not understood until around 2006.
In this talk, first we formulate the turbulization along a closed transversal. Then in our dimension setting, namely 2-dimensional foliations on 4-manifolds ((4,2)-foliations), a cohomological criterion is given for a given transversal to a foliation, which tells the turbulization is possible or not, relying on the Thurston's h-principle. Also we give cocrete geometric constructions of turbulizations.
The cohomological criterion for turbulization is deduced from a more general criterion for a given embedded surface to be a compact leaf or a closed transversal of some foliation, which is stated in terms of the euler classes of tangent and normal bndle of the foliation to look for. The anormalous cohomological solutions for certain cases suggested the geometric realization of turbulization, while the cohomological criterion is obtained by the h-principle.
Some other modifications are also formulated for (4,2)-foliations and their possibility are assured by the anormalous solutions mentioned above. For some of them, good geometric realizations are not yet known. So far the difficulty lies on the problem of the connected components of the space of representations of the surface groups to Diff S^1.
If the time permits, some special features on the h-principle for 2-dimensional foliations are also explained.
2018年06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
松井 宏樹 氏 (千葉大学)
Topological full groups and generalizations of the Higman-Thompson groups (JAPANESE)
Tea: Common Room 16:30-17:00
松井 宏樹 氏 (千葉大学)
Topological full groups and generalizations of the Higman-Thompson groups (JAPANESE)
[ 講演概要 ]
For a topological dynamical system on the Cantor set, one can introduce its topological full group, which is a countable subgroup of the homeomorphism group of the Cantor set. The Higman-Thompson group V_n is regarded as the topological full group of the one-sided full shift over n symbols. Replacing the one-sided full shift with other dynamical systems, we obtain variants of the Higman-Thompson group. It is then natural to ask whether those generalized Higman-Thompson groups possess similar (or different) features. I would like to discuss isomorphism classes of these groups, finiteness properties, abelianizations, connections to C*-algebras and their K-theory, and so on.
For a topological dynamical system on the Cantor set, one can introduce its topological full group, which is a countable subgroup of the homeomorphism group of the Cantor set. The Higman-Thompson group V_n is regarded as the topological full group of the one-sided full shift over n symbols. Replacing the one-sided full shift with other dynamical systems, we obtain variants of the Higman-Thompson group. It is then natural to ask whether those generalized Higman-Thompson groups possess similar (or different) features. I would like to discuss isomorphism classes of these groups, finiteness properties, abelianizations, connections to C*-algebras and their K-theory, and so on.
2018年05月29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
佐藤 光樹 氏 (東京大学大学院数理科学研究科)
A partial order on nu+ equivalence classes (JAPANESE)
Tea: Common Room 16:30-17:00
佐藤 光樹 氏 (東京大学大学院数理科学研究科)
A partial order on nu+ equivalence classes (JAPANESE)
[ 講演概要 ]
The nu+ equivalence is an equivalence relation on the knot concordance group. Hom proves that many concordance invariants derived from Heegaard Floer homology are invariant under nu+ equivalence. In this work, we introduce a partial order on nu+ equivalence classes, and study its algebraic and geometrical properties. As an application, we prove that any genus one knot is nu+ equivalent to one of the unknot, the trefoil and its mirror.
The nu+ equivalence is an equivalence relation on the knot concordance group. Hom proves that many concordance invariants derived from Heegaard Floer homology are invariant under nu+ equivalence. In this work, we introduce a partial order on nu+ equivalence classes, and study its algebraic and geometrical properties. As an application, we prove that any genus one knot is nu+ equivalent to one of the unknot, the trefoil and its mirror.
2018年05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
高田 土満 氏 (東京大学大学院数理科学研究科)
無限次元多様体の解析的指数とKK理論 (JAPANESE)
Tea: Common Room 16:30-17:00
高田 土満 氏 (東京大学大学院数理科学研究科)
無限次元多様体の解析的指数とKK理論 (JAPANESE)
[ 講演概要 ]
Atiyah-Singerの指数定理は，閉多様体上の解析的指数と位相的指数が一致することを主張する，微分トポロジーの金字塔の一つである．私の研究目標は，その指数理論の無限次元多様体版を与えることである．そのためには，できるだけ単純な場合から始めるのが自然であるため，次の問題を考えることにした：円周Tのループ群LTが，「固有かつ余コンパクトに」作用している無限次元多様体に対するLT同変指数理論を，KK理論的な観点から構築せよ．いまだにこの問題の解決には至っていないが，arXiv:1701.06055，arXiv:1709.06205 では，「関数空間」と見なせるHilbert空間を始めとする，解析的指数理論を構築するのに不可欠な対象をいくつか構成した．本講演では，この問題に対する現時点での結果を説明する．
Atiyah-Singerの指数定理は，閉多様体上の解析的指数と位相的指数が一致することを主張する，微分トポロジーの金字塔の一つである．私の研究目標は，その指数理論の無限次元多様体版を与えることである．そのためには，できるだけ単純な場合から始めるのが自然であるため，次の問題を考えることにした：円周Tのループ群LTが，「固有かつ余コンパクトに」作用している無限次元多様体に対するLT同変指数理論を，KK理論的な観点から構築せよ．いまだにこの問題の解決には至っていないが，arXiv:1701.06055，arXiv:1709.06205 では，「関数空間」と見なせるHilbert空間を始めとする，解析的指数理論を構築するのに不可欠な対象をいくつか構成した．本講演では，この問題に対する現時点での結果を説明する．
2018年05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
岡 睦雄 氏 (東京理科大学)
超曲面混合特異点理論とある予想 (JAPANESE)
Tea: Common Room 16:30-17:00
岡 睦雄 氏 (東京理科大学)
超曲面混合特異点理論とある予想 (JAPANESE)
[ 講演概要 ]
Consider a real algebraic variety of real codimension 2 defined by $V:=\{g(\mathbf x,\mathbf y)=h(\mathbf x,\mathbf y)=0\}$ in $\mathbb C^n=\mathbb R^n\times \mathbb R^n$. Put $\mathbf z=\mathbf x+i\mathbf y$ and consider complex valued real analytic function $f=g+ih$. Replace the variables $x_1,y_1\dots, x_n,y_n$ using the equality $x_j=(z_j+\bar z_j)/2,\, y_j=(z_j-\bar z_j)/2i$. Then $f$ can be understood to be an analytic functions of $z_j,\bar z_j$. We call $f$ a mixed function. In this way, $V=\{f(\mathbf z,\bar{\mathbf z})=0\}$ and we can use the techniques of complex analytic functions and the singularity theory developed there. In this talk, we explain basic properties of the singularity of mixed hyper surface $V(f)$ and give several open questions.
Consider a real algebraic variety of real codimension 2 defined by $V:=\{g(\mathbf x,\mathbf y)=h(\mathbf x,\mathbf y)=0\}$ in $\mathbb C^n=\mathbb R^n\times \mathbb R^n$. Put $\mathbf z=\mathbf x+i\mathbf y$ and consider complex valued real analytic function $f=g+ih$. Replace the variables $x_1,y_1\dots, x_n,y_n$ using the equality $x_j=(z_j+\bar z_j)/2,\, y_j=(z_j-\bar z_j)/2i$. Then $f$ can be understood to be an analytic functions of $z_j,\bar z_j$. We call $f$ a mixed function. In this way, $V=\{f(\mathbf z,\bar{\mathbf z})=0\}$ and we can use the techniques of complex analytic functions and the singularity theory developed there. In this talk, we explain basic properties of the singularity of mixed hyper surface $V(f)$ and give several open questions.
2018年05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
Dan Cristofaro-Gardiner 氏 (University of California, Santa Cruz)
Beyond the Weinstein conjecture (ENGLISH)
Tea: Common Room 16:30-17:00
Dan Cristofaro-Gardiner 氏 (University of California, Santa Cruz)
Beyond the Weinstein conjecture (ENGLISH)
[ 講演概要 ]
The Weinstein conjecture states that any Reeb vector field on a closed manifold has at least one closed orbit. The three-dimensional case of this conjecture was proved by Taubes in 2007, and Hutchings and I later showed that in this case there are always at least 2 orbits. While examples exist with exactly two orbits, one expects that this lower bound can be significantly improved with additional assumptions. For example, a theorem of Hofer, Wysocki, and Zehnder states that a generic nondegenerate Reeb vector field associated to the "standard" contact structure on $S^3$ has either 2, or infinitely many, closed orbits. We prove that any nondegenerate Reeb vector field has 2 or infinitely many closed orbits as long as the associated contact structure has torsion first Chern class. This is joint work with Mike Hutchings and Dan Pomerleano.
The Weinstein conjecture states that any Reeb vector field on a closed manifold has at least one closed orbit. The three-dimensional case of this conjecture was proved by Taubes in 2007, and Hutchings and I later showed that in this case there are always at least 2 orbits. While examples exist with exactly two orbits, one expects that this lower bound can be significantly improved with additional assumptions. For example, a theorem of Hofer, Wysocki, and Zehnder states that a generic nondegenerate Reeb vector field associated to the "standard" contact structure on $S^3$ has either 2, or infinitely many, closed orbits. We prove that any nondegenerate Reeb vector field has 2 or infinitely many closed orbits as long as the associated contact structure has torsion first Chern class. This is joint work with Mike Hutchings and Dan Pomerleano.
2018年04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
山本 卓宏 氏 (東京学芸大学)
Singular Fibers of smooth maps and Cobordism groups (JAPANESE)
Tea: Common Room 16:30-17:00
山本 卓宏 氏 (東京学芸大学)
Singular Fibers of smooth maps and Cobordism groups (JAPANESE)
[ 講演概要 ]
Following the pioneering work of R.Thom, cobordism groups of smooth maps have been studied by some mathematicians. Especially, cobordism groups of Morse functions on closed manifolds was studied by O.Saeki and K.Ikegami, B.Kalmar. In this talk, we will introduce cobordism groups among Morse functions on compact manifolds with boundary and study the cobordism groups are trivial or not by using the theory of the universal complex of singular fibers of smooth maps.
Following the pioneering work of R.Thom, cobordism groups of smooth maps have been studied by some mathematicians. Especially, cobordism groups of Morse functions on closed manifolds was studied by O.Saeki and K.Ikegami, B.Kalmar. In this talk, we will introduce cobordism groups among Morse functions on compact manifolds with boundary and study the cobordism groups are trivial or not by using the theory of the universal complex of singular fibers of smooth maps.
2018年04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
Tamás Kálmán 氏 (東京工業大学)
Tight contact structures on Seifert surface complements and knot invariants (ENGLISH)
Tea: Common Room 16:30-17:00
Tamás Kálmán 氏 (東京工業大学)
Tight contact structures on Seifert surface complements and knot invariants (ENGLISH)
[ 講演概要 ]
In joint work with Daniel Mathews, we examined complements of standard Seifert surfaces of special alternating links and used Honda's method to enumerate those tight contact structures on them whose dividing sets are isotopic to the link. The number turns out to be the leading coefficient of the Alexander polynomial. The proof is rather combinatorial in nature; for example, the Euler classes of the contact structures are identified with `hypertrees' in a certain hypergraph. Using earlier results with Hitoshi Murakami and Alexander Postnikov, this yields a connection between contact topology and the Homfly polynomial. We also found that the contact invariants of our structures form a basis for the sutured Floer homology of the manifold.
In joint work with Daniel Mathews, we examined complements of standard Seifert surfaces of special alternating links and used Honda's method to enumerate those tight contact structures on them whose dividing sets are isotopic to the link. The number turns out to be the leading coefficient of the Alexander polynomial. The proof is rather combinatorial in nature; for example, the Euler classes of the contact structures are identified with `hypertrees' in a certain hypergraph. Using earlier results with Hitoshi Murakami and Alexander Postnikov, this yields a connection between contact topology and the Homfly polynomial. We also found that the contact invariants of our structures form a basis for the sutured Floer homology of the manifold.
2018年04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
遠藤 久顕 氏 (東京工業大学)
２次元結び目のモース・ノビコフ数について (JAPANESE)
Tea: Common Room 16:30-17:00
遠藤 久顕 氏 (東京工業大学)
２次元結び目のモース・ノビコフ数について (JAPANESE)
[ 講演概要 ]
2001年にPajitnov, Rudolph, Weberは，古典的絡み目に対してMorse-Novikov数を定義し，その基本的な性質を研究した．この不変量は，Alexander多項式やNovikovホモロジーおよびそれらの「ねじれ版」との関連から盛んに研究されている．本講演では，２次元結び目に対してMorse-Novikov数を定義し，いくつかの性質を述べる．特に，２次元結び目のモーション・ピクチャーやスピン構成法との関係について解説する．尚，本講演の内容はAndrei Pajitnov氏（ナント大学）との共同研究にもとづいている．
2001年にPajitnov, Rudolph, Weberは，古典的絡み目に対してMorse-Novikov数を定義し，その基本的な性質を研究した．この不変量は，Alexander多項式やNovikovホモロジーおよびそれらの「ねじれ版」との関連から盛んに研究されている．本講演では，２次元結び目に対してMorse-Novikov数を定義し，いくつかの性質を述べる．特に，２次元結び目のモーション・ピクチャーやスピン構成法との関係について解説する．尚，本講演の内容はAndrei Pajitnov氏（ナント大学）との共同研究にもとづいている．
2018年04月03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
入江 慶 氏 (東京大学大学院数理科学研究科)
Chain level loop bracket and pseudo-holomorphic disks (JAPANESE)
Tea: Common Room 16:30-17:00
入江 慶 氏 (東京大学大学院数理科学研究科)
Chain level loop bracket and pseudo-holomorphic disks (JAPANESE)
[ 講演概要 ]
Let $L$ be a Lagrangian submanifold in a symplectic vector space which is closed, oriented and spin. Using virtual fundamental chains of moduli spaces of nonconstant pseudo-holomorphic disks with boundaries on $L$, one can define a Maurer-Cartan element of a Lie bracket operation in string topology (the loop bracket) defined at chain level. This idea is due to Fukaya, who also pointed out its important consequences in symplectic topology. In this talk I will explain how to rigorously carry out this idea. Our argument is based on a string topology chain model previously introduced by the speaker, and theory of Kuranishi structures on moduli spaces of pseudo-holomorphic disks, which has been developed by Fukaya-Oh-Ohta-Ono.
Let $L$ be a Lagrangian submanifold in a symplectic vector space which is closed, oriented and spin. Using virtual fundamental chains of moduli spaces of nonconstant pseudo-holomorphic disks with boundaries on $L$, one can define a Maurer-Cartan element of a Lie bracket operation in string topology (the loop bracket) defined at chain level. This idea is due to Fukaya, who also pointed out its important consequences in symplectic topology. In this talk I will explain how to rigorously carry out this idea. Our argument is based on a string topology chain model previously introduced by the speaker, and theory of Kuranishi structures on moduli spaces of pseudo-holomorphic disks, which has been developed by Fukaya-Oh-Ohta-Ono.
2018年03月30日(金)
15:00-16:30 数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
Matteo Felder 氏 (University of Geneva)
Graph Complexes and the Kashiwara-Vergne Lie algebra (ENGLISH)
Tea: Common Room 16:30-17:00
Matteo Felder 氏 (University of Geneva)
Graph Complexes and the Kashiwara-Vergne Lie algebra (ENGLISH)
[ 講演概要 ]
The Kashiwara-Vergne Lie algebra krv was introduced by A. Alekseev and C. Torossian. It describes the symmetries of the Kashiwara-Vergne problem in Lie theory. It has been shown to contain the Grothendieck-Teichmüller Lie algebra grt as a Lie subalgebra. Conjecturally, these two Lie algebras are expected to be isomorphic. An important theorem by T. Willwacher states that in degree zero the cohomology of M. Kontsevich's graph complex GC is isomorphic to grt. We will show how T. Willwacher's result induces a natural way to define a nested sequence of Lie subalgebras of krv whose intersection is grt. This infinite family therefore interpolates between the two Lie algebras. For this we will recall several techniques from the theory of graph complexes. If time permits, we will then sketch how one might generalize these notions to establish a "genus one" analogue of T. Willwacher theorem. More precisely, we will define a chain complex whose degree zero cohomology is given by a Lie subalgebra of the elliptic Grothendieck-Teichmüller Lie algebra introduced by B. Enriquez. The last part is joint work in progress with T. Willwacher.
The Kashiwara-Vergne Lie algebra krv was introduced by A. Alekseev and C. Torossian. It describes the symmetries of the Kashiwara-Vergne problem in Lie theory. It has been shown to contain the Grothendieck-Teichmüller Lie algebra grt as a Lie subalgebra. Conjecturally, these two Lie algebras are expected to be isomorphic. An important theorem by T. Willwacher states that in degree zero the cohomology of M. Kontsevich's graph complex GC is isomorphic to grt. We will show how T. Willwacher's result induces a natural way to define a nested sequence of Lie subalgebras of krv whose intersection is grt. This infinite family therefore interpolates between the two Lie algebras. For this we will recall several techniques from the theory of graph complexes. If time permits, we will then sketch how one might generalize these notions to establish a "genus one" analogue of T. Willwacher theorem. More precisely, we will define a chain complex whose degree zero cohomology is given by a Lie subalgebra of the elliptic Grothendieck-Teichmüller Lie algebra introduced by B. Enriquez. The last part is joint work in progress with T. Willwacher.
2018年03月30日(金)
17:00-18:30 数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
Florian Naef 氏 (Massachusetts Institute of Technology)
Goldman-Turaev formality in genus 0 from the KZ connection (ENGLISH)
Tea: Common Room 16:30-17:00
Florian Naef 氏 (Massachusetts Institute of Technology)
Goldman-Turaev formality in genus 0 from the KZ connection (ENGLISH)
[ 講演概要 ]
Using the intersection and self-intersection of loops on a surface one can define the Goldman-Turaev Lie bialgebra. By earlier joint work with A. Alekseev, N. Kawazumi and Y. Kuno it is known that the linearization problem of the Goldman-Turaev Lie bialgebra is closely related to the Kashiwara-Vergne problem and hence to Drinfeld associators. It turns out that in the case of a genus 0 surface and over the field of complex numbers there is a very direct and explicit proof of the formality of the Goldman-Turaev Lie bialgebra using the monodromy of the Knizhnik-Zamolodchikov connection. This is joint work with Anton Alekseev.
Using the intersection and self-intersection of loops on a surface one can define the Goldman-Turaev Lie bialgebra. By earlier joint work with A. Alekseev, N. Kawazumi and Y. Kuno it is known that the linearization problem of the Goldman-Turaev Lie bialgebra is closely related to the Kashiwara-Vergne problem and hence to Drinfeld associators. It turns out that in the case of a genus 0 surface and over the field of complex numbers there is a very direct and explicit proof of the formality of the Goldman-Turaev Lie bialgebra using the monodromy of the Knizhnik-Zamolodchikov connection. This is joint work with Anton Alekseev.
2018年02月21日(水)
17:00-18:30 数理科学研究科棟(駒場) 122号室
Tea: Common Room 16:30-17:00
Gwénaël Massuyeau 氏 (Université de Bourgogne)
The category of bottom tangles in handlebodies, and the Kontsevich integral (ENGLISH)
Tea: Common Room 16:30-17:00
Gwénaël Massuyeau 氏 (Université de Bourgogne)
The category of bottom tangles in handlebodies, and the Kontsevich integral (ENGLISH)
[ 講演概要 ]
Habiro introduced the category B of « bottom tangles in handlebodies », which encapsulates the set of knots in the 3-sphere as well as the mapping class groups of 3-dimensional handlebodies. There is a natural filtration on the category B defined using an appropriate generalization of Vassiliev invariants. In this talk, we will show that the completion of B with respect to the Vassiliev filtration is isomorphic to a certain category A which can be defined either in a combinatorial way using « Jacobi diagrams », or by a universal property via the notion of « Casimir Hopf algebra ». Such an isomorphism will be obtained by extending the Kontsevich integral (originally defined as a knot invariant) to a functor Z from B to A. This functor Z can be regarded as a refinement of the TQFT-like functor derived from the LMO invariant and, if time allows, we will evoke the topological interpretation of the « tree-level » of Z. (This is based on joint works with Kazuo Habiro.)
Habiro introduced the category B of « bottom tangles in handlebodies », which encapsulates the set of knots in the 3-sphere as well as the mapping class groups of 3-dimensional handlebodies. There is a natural filtration on the category B defined using an appropriate generalization of Vassiliev invariants. In this talk, we will show that the completion of B with respect to the Vassiliev filtration is isomorphic to a certain category A which can be defined either in a combinatorial way using « Jacobi diagrams », or by a universal property via the notion of « Casimir Hopf algebra ». Such an isomorphism will be obtained by extending the Kontsevich integral (originally defined as a knot invariant) to a functor Z from B to A. This functor Z can be regarded as a refinement of the TQFT-like functor derived from the LMO invariant and, if time allows, we will evoke the topological interpretation of the « tree-level » of Z. (This is based on joint works with Kazuo Habiro.)
2018年01月30日(火)
17:00-18:00 数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
池 祐一 氏 (東京大学大学院数理科学研究科)
Persistence-like distance on Tamarkin's category and symplectic displacement energy (JAPANESE)
Tea: Common Room 16:30-17:00
池 祐一 氏 (東京大学大学院数理科学研究科)
Persistence-like distance on Tamarkin's category and symplectic displacement energy (JAPANESE)
[ 講演概要 ]
The microlocal sheaf theory due to Kashiwara and Schapira can be regarded as Morse theory with sheaf coefficients. Recently it has been applied to symplectic geometry, after the pioneering work of Tamarkin. In this talk, I will propose a new sheaf-theoretic method to estimate the displacement energy of compact subsets in cotangent bundles. In the course of the proof, we introduce a persistence-like pseudo-distance on Tamarkin's sheaf category. This is a joint work with Tomohiro Asano.
The microlocal sheaf theory due to Kashiwara and Schapira can be regarded as Morse theory with sheaf coefficients. Recently it has been applied to symplectic geometry, after the pioneering work of Tamarkin. In this talk, I will propose a new sheaf-theoretic method to estimate the displacement energy of compact subsets in cotangent bundles. In the course of the proof, we introduce a persistence-like pseudo-distance on Tamarkin's sheaf category. This is a joint work with Tomohiro Asano.
2018年01月23日(火)
17:00-18:00 数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
野崎 雄太 氏 (東京大学大学院数理科学研究科)
An invariant of 3-manifolds via homology cobordisms (JAPANESE)
Tea: Common Room 16:30-17:00
野崎 雄太 氏 (東京大学大学院数理科学研究科)
An invariant of 3-manifolds via homology cobordisms (JAPANESE)
[ 講演概要 ]
For a closed 3-manifold X, we consider the topological invariant defined as the minimal integer g such that X is obtained as the closure of a homology cobordism over a surface of genus g. We prove that the invariant equals one for every lens space, which is contrast to the fact that some lens spaces do not admit any open book decomposition whose page is a surface of genus one. The proof is based on the Chebotarev density theorem and binary quadratic forms in number theory.
For a closed 3-manifold X, we consider the topological invariant defined as the minimal integer g such that X is obtained as the closure of a homology cobordism over a surface of genus g. We prove that the invariant equals one for every lens space, which is contrast to the fact that some lens spaces do not admit any open book decomposition whose page is a surface of genus one. The proof is based on the Chebotarev density theorem and binary quadratic forms in number theory.
2018年01月23日(火)
18:00-19:00 数理科学研究科棟(駒場) 056号室
田中 淳波 氏 (東京大学大学院数理科学研究科)
Wrapping projections and decompositions of Keinian groups (JAPANESE)
田中 淳波 氏 (東京大学大学院数理科学研究科)
Wrapping projections and decompositions of Keinian groups (JAPANESE)
[ 講演概要 ]
Let $S$ be a closed surface of genus $g ¥geq 2$. The deformation space $AH(S)$ consists of (conjugacy classes of) discrete faithful representations $\rho:\pi_{1}(S) \to PSL_{2}(\mathbb{C})$.
McMullen, and Bromberg and Holt showed that $AH(S)$ can self-bump, that is, the interior of $AH(S)$ has the self-intersecting closure.
Both of them demonstrated the existence of self-bumping under the exisetence of a non-trivial wrapping projections from an algebraic limits to a geometric limits which wraps an annulus cusp into a torus cusp.
In this talk, given a representation $\rho$ at the boundary of $AH(S)$, we characterize a wrapping projection to a geometric limit associated to $\rho$, by the information of the actions of decomposed Kleinian groups of the image of $\rho$.
Let $S$ be a closed surface of genus $g ¥geq 2$. The deformation space $AH(S)$ consists of (conjugacy classes of) discrete faithful representations $\rho:\pi_{1}(S) \to PSL_{2}(\mathbb{C})$.
McMullen, and Bromberg and Holt showed that $AH(S)$ can self-bump, that is, the interior of $AH(S)$ has the self-intersecting closure.
Both of them demonstrated the existence of self-bumping under the exisetence of a non-trivial wrapping projections from an algebraic limits to a geometric limits which wraps an annulus cusp into a torus cusp.
In this talk, given a representation $\rho$ at the boundary of $AH(S)$, we characterize a wrapping projection to a geometric limit associated to $\rho$, by the information of the actions of decomposed Kleinian groups of the image of $\rho$.
2018年01月16日(火)
17:00-18:00 数理科学研究科棟(駒場) 056号室
Tea: Common Room 16:30-17:00
Jimenez Pascual Adrian 氏 (東京大学大学院数理科学研究科)
On adequacy and the crossing number of satellite knots (JAPANESE)
Tea: Common Room 16:30-17:00
Jimenez Pascual Adrian 氏 (東京大学大学院数理科学研究科)
On adequacy and the crossing number of satellite knots (JAPANESE)
[ 講演概要 ]
It has always been difficult to prove results regarding the (minimal) crossing number of knots. In particular, apparently easy problems such as knowing the crossing number of the connected sum of knots, or bounding the crossing number of satellite knots have been conjectured through decades, yet still remain open. Focusing on this latter problem, in this talk I will prove that the crossing number of a satellite knot is bounded from below by the crossing number of its companion, when the companion is adequate.
It has always been difficult to prove results regarding the (minimal) crossing number of knots. In particular, apparently easy problems such as knowing the crossing number of the connected sum of knots, or bounding the crossing number of satellite knots have been conjectured through decades, yet still remain open. Focusing on this latter problem, in this talk I will prove that the crossing number of a satellite knot is bounded from below by the crossing number of its companion, when the companion is adequate.
