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トポロジー火曜セミナー
過去の記録 ～05/21｜次回の予定｜今後の予定 05/22～
| 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, 葉廣和夫 |
| セミナーURL | https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index.html |
過去の記録
2025年05月13日(火)
17:00-18:30 数理科学研究科棟(駒場) ハイブリッド開催/056号室
対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
池 祐一 氏 (東京大学大学院数理科学研究科)
Interleaving distance for sheaves and its application to symplectic geometry (JAPANESE)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
池 祐一 氏 (東京大学大学院数理科学研究科)
Interleaving distance for sheaves and its application to symplectic geometry (JAPANESE)
[ 講演概要 ]
The Interleaving distance was first introduced in the context of the stability of persistent homology and is now used in various fields. It was adapted to sheaves by the pioneering work of Curry, and later in the derived setting by Kashiwara and Schapira. In this talk, I will explain that the interleaving distance for sheaves is related to the energy of Hamiltonian actions on cotangent bundles. Moreover, I will show that the derived interleaving distance is complete, which enables us to treat non-smooth objects in symplectic geometry using sheaf-theoretic methods. This is based on joint work with Tomohiro Asano, Stéphane Guillermou, Vincent Humilière, and Claude Viterbo.
[ 参考URL ]The Interleaving distance was first introduced in the context of the stability of persistent homology and is now used in various fields. It was adapted to sheaves by the pioneering work of Curry, and later in the derived setting by Kashiwara and Schapira. In this talk, I will explain that the interleaving distance for sheaves is related to the energy of Hamiltonian actions on cotangent bundles. Moreover, I will show that the derived interleaving distance is complete, which enables us to treat non-smooth objects in symplectic geometry using sheaf-theoretic methods. This is based on joint work with Tomohiro Asano, Stéphane Guillermou, Vincent Humilière, and Claude Viterbo.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
2025年04月22日(火)
17:30-18:30 数理科学研究科棟(駒場) hybrid/056号室
Lie 群論・表現論セミナーと合同。 参加を希望される場合は、セミナーのウェブページをご覧下さい。
奥田 隆幸 氏 (広島大学)
Coarse coding theory and discontinuous groups on homogeneous spaces (JAPANESE)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
Lie 群論・表現論セミナーと合同。 参加を希望される場合は、セミナーのウェブページをご覧下さい。
奥田 隆幸 氏 (広島大学)
Coarse coding theory and discontinuous groups on homogeneous spaces (JAPANESE)
[ 講演概要 ]
Let $M$ and $\mathcal{I}$ be sets, and consider a surjective map
\[ R : M \times M \to \mathcal{I}. \]
For each subset $\mathcal{A} \subseteq \mathcal{I}$, we define $\mathcal{A}$-free codes on $M$ as subsets $C \subseteq M$ satisfying
\[ R(C \times C) \cap \mathcal{A} = \emptyset. \]
This definition encompasses various types of codes, including error-correcting codes, spherical codes, and those defined on association schemes or homogeneous spaces. In this talk, we introduce a "pre-bornological coarse structure" on $\mathcal{I}$ and define the notion of coarsely $\mathcal{A}$-free codes on $M$. This extends the concept of $\mathcal{A}$-free codes introduced above. As a main result, we establish relationships between coarse coding theory on Riemannian homogeneous spaces $M = G/K$ and discontinuous group theory on non-Riemannian homogeneous spaces $X = G/H$.
[ 参考URL ]Let $M$ and $\mathcal{I}$ be sets, and consider a surjective map
\[ R : M \times M \to \mathcal{I}. \]
For each subset $\mathcal{A} \subseteq \mathcal{I}$, we define $\mathcal{A}$-free codes on $M$ as subsets $C \subseteq M$ satisfying
\[ R(C \times C) \cap \mathcal{A} = \emptyset. \]
This definition encompasses various types of codes, including error-correcting codes, spherical codes, and those defined on association schemes or homogeneous spaces. In this talk, we introduce a "pre-bornological coarse structure" on $\mathcal{I}$ and define the notion of coarsely $\mathcal{A}$-free codes on $M$. This extends the concept of $\mathcal{A}$-free codes introduced above. As a main result, we establish relationships between coarse coding theory on Riemannian homogeneous spaces $M = G/K$ and discontinuous group theory on non-Riemannian homogeneous spaces $X = G/H$.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
2025年04月15日(火)
17:00-18:30 数理科学研究科棟(駒場) hybrid/056号室
対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
坂井 健人 氏 (東京大学大学院数理科学研究科)
Harmonic maps and uniform degeneration of hyperbolic surfaces with boundary (JAPANESE)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
坂井 健人 氏 (東京大学大学院数理科学研究科)
Harmonic maps and uniform degeneration of hyperbolic surfaces with boundary (JAPANESE)
[ 講演概要 ]
If holomorphic quadratic differentials on a punctured Riemann surface have poles of order >1 at the punctures, they correspond to hyperbolic surfaces with geodesic boundary via harmonic maps. This correspondence is known as the harmonic map parametrization of hyperbolic surfaces. In this talk, we use this parametrization to describe the degeneration of hyperbolic surfaces via Gromov-Hausdorff convergence. As an application, we study the limit of a one-parameter family of hyperbolic surfaces in the Thurston boundary of Teichmüller space.
[ 参考URL ]If holomorphic quadratic differentials on a punctured Riemann surface have poles of order >1 at the punctures, they correspond to hyperbolic surfaces with geodesic boundary via harmonic maps. This correspondence is known as the harmonic map parametrization of hyperbolic surfaces. In this talk, we use this parametrization to describe the degeneration of hyperbolic surfaces via Gromov-Hausdorff convergence. As an application, we study the limit of a one-parameter family of hyperbolic surfaces in the Thurston boundary of Teichmüller space.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
2025年04月08日(火)
17:00-18:30 数理科学研究科棟(駒場) hybrid/056号室
対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
高津 飛鳥 氏 (東京大学大学院数理科学研究科)
Concavity and Dirichlet heat flow (JAPANESE)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
高津 飛鳥 氏 (東京大学大学院数理科学研究科)
Concavity and Dirichlet heat flow (JAPANESE)
[ 講演概要 ]
In a convex domain of Euclidean space, the Dirichlet heat flow transmits log-concavity from the initial time to any time. I first introduce a notion of generalized concavity and specify a concavity preserved by the Dirichlet heat flow. Then I show that in a totally convex domain of a Riemannian manifold, if some concavity is preserved by the Dirichlet heat flow, then the sectional curvature must vanish on the domain. The first part is based on joint work with Kazuhiro Ishige and Paolo Salani, and the second part is based on joint work with Kazuhiro Ishige and Haruto Tokunaga.
[ 参考URL ]In a convex domain of Euclidean space, the Dirichlet heat flow transmits log-concavity from the initial time to any time. I first introduce a notion of generalized concavity and specify a concavity preserved by the Dirichlet heat flow. Then I show that in a totally convex domain of a Riemannian manifold, if some concavity is preserved by the Dirichlet heat flow, then the sectional curvature must vanish on the domain. The first part is based on joint work with Kazuhiro Ishige and Paolo Salani, and the second part is based on joint work with Kazuhiro Ishige and Haruto Tokunaga.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
2025年01月21日(火)
17:00-18:00 数理科学研究科棟(駒場) hybrid/056号室
対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
小菅 亮太朗 氏 (東京大学大学院数理科学研究科)
Rational abelianizations of Chillingworth subgroups of mapping class groups and automorphism groups of free groups (JAPANESE)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
小菅 亮太朗 氏 (東京大学大学院数理科学研究科)
Rational abelianizations of Chillingworth subgroups of mapping class groups and automorphism groups of free groups (JAPANESE)
[ 講演概要 ]
The Chillingworth subgroup of the mapping class group of a surface is defined as the subgroup consisting of elements that preserve nonsingular vector fields up to homotopy. The action of the mapping class group on the set of homotopy classes of nonsingular vector fields is described using the concept of the winding number. By employing a cohomological approach, we extend the notion of the winding number to general manifolds, introducing the definition of the Chillingworth subgroup for both the mapping class group of general manifolds and the automorphism group of a free group. In this work, we determine the rational abelianization of the Chillingworth subgroup of the mapping class group of a surface and, under a certain assumption, also determine the rational abelianization of the Chillingworth subgroup for the automorphism group of a free group.
[ 参考URL ]The Chillingworth subgroup of the mapping class group of a surface is defined as the subgroup consisting of elements that preserve nonsingular vector fields up to homotopy. The action of the mapping class group on the set of homotopy classes of nonsingular vector fields is described using the concept of the winding number. By employing a cohomological approach, we extend the notion of the winding number to general manifolds, introducing the definition of the Chillingworth subgroup for both the mapping class group of general manifolds and the automorphism group of a free group. In this work, we determine the rational abelianization of the Chillingworth subgroup of the mapping class group of a surface and, under a certain assumption, also determine the rational abelianization of the Chillingworth subgroup for the automorphism group of a free group.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
2025年01月14日(火)
17:00-18:00 数理科学研究科棟(駒場) hybrid/056号室
対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
吉岡 玲音 氏 (東京大学大学院数理科学研究科)
Some non-trivial cycles of the space of long embeddings detected by configuration space integral invariants using g-loop graphs (JAPANESE)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
吉岡 玲音 氏 (東京大学大学院数理科学研究科)
Some non-trivial cycles of the space of long embeddings detected by configuration space integral invariants using g-loop graphs (JAPANESE)
[ 講演概要 ]
In this talk, we give some non-trivial cocycles and cycles of the space of long embeddings R^j --> R^n modulo immersions. First, we construct a cocycle through configuration space integrals with the simplest 2-loop graph cocycle of the Bott-Cattaneo-Rossi graph complex for odd n and j. Then, we give a cycle from a chord diagram on oriented lines, which is associated with the simplest 2-loop hairy graph. We show the non-triviality of this (co)cycle by pairing argument, which is reduced to pairing of graphs with the chord diagram. This construction of cycles and the pairing argument to show the non-triviality is also applied to general 2-loop (co)cycles of top degree. If time permits, we introduce a modified graph complex and configuration space integrals to give more general cocycles. This new graph complex is quasi-isomorphic to both the hairy graph complex and the graph complex introduced in embedding calculus by Arone and Turchin. With these modified cocycles, our pairing argument provides an alternative proof of the non-finite generation of the (j-1)-th rational homotopy group of the space of long j-knots R^j -->R^{j+2}, which Budney-Gabai and Watanabe first established.
[ 参考URL ]In this talk, we give some non-trivial cocycles and cycles of the space of long embeddings R^j --> R^n modulo immersions. First, we construct a cocycle through configuration space integrals with the simplest 2-loop graph cocycle of the Bott-Cattaneo-Rossi graph complex for odd n and j. Then, we give a cycle from a chord diagram on oriented lines, which is associated with the simplest 2-loop hairy graph. We show the non-triviality of this (co)cycle by pairing argument, which is reduced to pairing of graphs with the chord diagram. This construction of cycles and the pairing argument to show the non-triviality is also applied to general 2-loop (co)cycles of top degree. If time permits, we introduce a modified graph complex and configuration space integrals to give more general cocycles. This new graph complex is quasi-isomorphic to both the hairy graph complex and the graph complex introduced in embedding calculus by Arone and Turchin. With these modified cocycles, our pairing argument provides an alternative proof of the non-finite generation of the (j-1)-th rational homotopy group of the space of long j-knots R^j -->R^{j+2}, which Budney-Gabai and Watanabe first established.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
2024年12月17日(火)
17:00-18:30 数理科学研究科棟(駒場) hybrid/056号室
対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
Emmanuel Graff 氏 (東京大学大学院数理科学研究科)
Is there torsion in the homotopy braid group? (ENGLISH)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
Emmanuel Graff 氏 (東京大学大学院数理科学研究科)
Is there torsion in the homotopy braid group? (ENGLISH)
[ 講演概要 ]
In the 'Kourovka notebook,' V. Lin questions the existence of a non-trivial epimorphism from the braid group onto a non-abelian torsion-free group. The homotopy braid group, studied by Goldsmith in 1974, naturally appears as a potential candidate. In 2001, Humphries showed that this homotopy braid group is torsion-free for less than six strands. In this presentation, we will see a new approach based on the broader concept of welded braids, along with algebraic techniques, to determine whether the homotopy braid group provides a complete answer to Lin’s question.
[ 参考URL ]In the 'Kourovka notebook,' V. Lin questions the existence of a non-trivial epimorphism from the braid group onto a non-abelian torsion-free group. The homotopy braid group, studied by Goldsmith in 1974, naturally appears as a potential candidate. In 2001, Humphries showed that this homotopy braid group is torsion-free for less than six strands. In this presentation, we will see a new approach based on the broader concept of welded braids, along with algebraic techniques, to determine whether the homotopy braid group provides a complete answer to Lin’s question.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
2024年12月10日(火)
17:00-18:30 数理科学研究科棟(駒場) ハイブリッド開催/056号室
対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
若月 駿 氏 (名古屋大学)
Computation of the magnitude homology as a derived functor (JAPANESE)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
若月 駿 氏 (名古屋大学)
Computation of the magnitude homology as a derived functor (JAPANESE)
[ 講演概要 ]
Asao-Ivanov showed that the magnitude homology of a finite metric space is isomorphic to the derived functor Tor over some ring. In this talk, I will explain an application of the theory of minimal projective resolution to this derived functor. Especially in the case of a geodetic graph, torsion-freeness and a criterion for diagonality of the magnitude homology are established. Moreover, I will give computational examples including cyclic graphs. This is a joint work with Yasuhiko Asao.
[ 参考URL ]Asao-Ivanov showed that the magnitude homology of a finite metric space is isomorphic to the derived functor Tor over some ring. In this talk, I will explain an application of the theory of minimal projective resolution to this derived functor. Especially in the case of a geodetic graph, torsion-freeness and a criterion for diagonality of the magnitude homology are established. Moreover, I will give computational examples including cyclic graphs. This is a joint work with Yasuhiko Asao.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
2024年12月03日(火)
17:30-18:30 数理科学研究科棟(駒場) hybrid/056号室
対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
井ノ口 順一 氏 (北海道大学)
３次元空間内の曲面と可積分系 (JAPANESE)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
井ノ口 順一 氏 (北海道大学)
３次元空間内の曲面と可積分系 (JAPANESE)
[ 講演概要 ]
次元双曲空間の平均曲率一定曲面は平均曲率の値により様相が異なる．とくに平均曲率の値が１未満の場合はユークリッド空間や球面に類似物をもたない双曲幾何特有のクラスを与える．本講演では平均曲率の値が１未満の平均曲率一定曲面の可積分系理論的構成法について解説する (Josef F. Dorfmeister氏, 小林真平氏との共同研究)．
[ 参考URL ]次元双曲空間の平均曲率一定曲面は平均曲率の値により様相が異なる．とくに平均曲率の値が１未満の場合はユークリッド空間や球面に類似物をもたない双曲幾何特有のクラスを与える．本講演では平均曲率の値が１未満の平均曲率一定曲面の可積分系理論的構成法について解説する (Josef F. Dorfmeister氏, 小林真平氏との共同研究)．
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
2024年11月26日(火)
17:00-18:30 数理科学研究科棟(駒場) hybrid/056号室
対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
名取 雅生 氏 (東京大学大学院数理科学研究科)
A proof of Bott periodicity via Quot schemes and bulk-edge correspondence (JAPANESE)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
名取 雅生 氏 (東京大学大学院数理科学研究科)
A proof of Bott periodicity via Quot schemes and bulk-edge correspondence (JAPANESE)
[ 講演概要 ]
The bulk-edge correspondence refers to the phenomenon typically found in topological insulators, where the topological restriction of the bulk (interior) determines the physical state, such as electric currents, at the edge (boundary). In this talk, we focus on the formulation by G. M. Graf and M. Porta and later by S. Hayashi and provide a new proof of bulk-edge correspondence. It is more direct compared to previous approaches. Behind the proof lies the Bott periodicity of K-theory. The proof of Bott periodicity has been approached from various perspectives. We provide a new proof of Bott periodicity. In the proof, we use Quot schemes in algebraic geometry as configuration spaces.
[ 参考URL ]The bulk-edge correspondence refers to the phenomenon typically found in topological insulators, where the topological restriction of the bulk (interior) determines the physical state, such as electric currents, at the edge (boundary). In this talk, we focus on the formulation by G. M. Graf and M. Porta and later by S. Hayashi and provide a new proof of bulk-edge correspondence. It is more direct compared to previous approaches. Behind the proof lies the Bott periodicity of K-theory. The proof of Bott periodicity has been approached from various perspectives. We provide a new proof of Bott periodicity. In the proof, we use Quot schemes in algebraic geometry as configuration spaces.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
2024年11月19日(火)
17:00-18:30 数理科学研究科棟(駒場) hybrid/056号室
対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
Bruno Scárdua 氏 (Federal University of Rio de Janeiro)
On real center singularities of complex vector fields on surfaces (ENGLISH)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
Bruno Scárdua 氏 (Federal University of Rio de Janeiro)
On real center singularities of complex vector fields on surfaces (ENGLISH)
[ 講演概要 ]
One of the various versions of the classical Lyapunov-Poincaré center theorem states that a nondegenerate real analytic center type planar vector field singularity admits an analytic first integral. In a more proof of this result, R. Moussu establishes important connection between this result and the theory of singularities of holomorphic foliations ([2]). In this paper we consider generalizations for two main frameworks: (i) planar real analytic vector fields with "many" periodic orbits near the singularity and
(ii) germs of holomorphic foliations having a suitable singularity in dimension two.
In this talk we discuss some versions of Poincaré-Lyapunov center theorem, including for the case of holomorphic vector fields. We also give some applications, hinting that there is much more to be explored in this framework.
References
[1] V. León, B. Scárdua, On a Theorem of Lyapunov-Poincaré in Higher Dimensions, July 2021, Arnold Mathematical Journal 7(3) DOI:10.1007/s40598-021-00183-x.
[2] R. Moussu: Une démonstration géométrique d’un théorème de Lyapunov-Poincaré. Astérisque, tome 98-99 (1982), p. 216-223.
[3] A. Lyapunov: Etude d’un cas particulier du problème de la stabilité du mouvement. Mat. Sbornik 17 (1893) pages 252-333 (Russe).
[4] H. Poincaré: Mémoire sur les courbes définies par une équation différentielle (I), Journal de mathématiques pures et appliquées 3e série, tome 7 (1881), p. 375-422.
[5] Minoru Urabe and Yasutaka Sibuya; On Center of Higher Dimensions; Journal of Science of the Hiroshima University, Ser. A, . Vol. 19, No. I, July, 1955.
[ 参考URL ]One of the various versions of the classical Lyapunov-Poincaré center theorem states that a nondegenerate real analytic center type planar vector field singularity admits an analytic first integral. In a more proof of this result, R. Moussu establishes important connection between this result and the theory of singularities of holomorphic foliations ([2]). In this paper we consider generalizations for two main frameworks: (i) planar real analytic vector fields with "many" periodic orbits near the singularity and
(ii) germs of holomorphic foliations having a suitable singularity in dimension two.
In this talk we discuss some versions of Poincaré-Lyapunov center theorem, including for the case of holomorphic vector fields. We also give some applications, hinting that there is much more to be explored in this framework.
References
[1] V. León, B. Scárdua, On a Theorem of Lyapunov-Poincaré in Higher Dimensions, July 2021, Arnold Mathematical Journal 7(3) DOI:10.1007/s40598-021-00183-x.
[2] R. Moussu: Une démonstration géométrique d’un théorème de Lyapunov-Poincaré. Astérisque, tome 98-99 (1982), p. 216-223.
[3] A. Lyapunov: Etude d’un cas particulier du problème de la stabilité du mouvement. Mat. Sbornik 17 (1893) pages 252-333 (Russe).
[4] H. Poincaré: Mémoire sur les courbes définies par une équation différentielle (I), Journal de mathématiques pures et appliquées 3e série, tome 7 (1881), p. 375-422.
[5] Minoru Urabe and Yasutaka Sibuya; On Center of Higher Dimensions; Journal of Science of the Hiroshima University, Ser. A, . Vol. 19, No. I, July, 1955.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
2024年11月12日(火)
17:30-18:30 数理科学研究科棟(駒場) ハイブリッド開催/056号室
Lie群論・表現論セミナーと合同開催。
井上 順子 氏 (鳥取大学)
Holomorphically induced representations of some solvable Lie groups (JAPANESE)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
Lie群論・表現論セミナーと合同開催。
井上 順子 氏 (鳥取大学)
Holomorphically induced representations of some solvable Lie groups (JAPANESE)
[ 講演概要 ]
From a viewpoint of the orbit method, holomorphic induction is originally based on the idea of realizing an irreducible unitary representation of a Lie group $G$ in an $L^2$-space of some holomorphic sections of some line bundle over a $G$-homogeneous space associated with a polarization for a linear form of the Lie algebra of $G$. It is a generalization of ordinary induction from a unitary character; Through this process, Auslander-Kostant constructed the irreducible unitary representations of type 1, connected, simply connected solvable Lie groups.
In this talk, focusing on the class of exponential solvable Lie groups, we are concerned with holomorphically induced representations $\rho$ in some general settings.
We would like to discuss the following problems:
(1) conditions of non-vanishing of $\rho$,
(2) decomposition of $\rho$ into a direct integral of irreducible representations,
(3) Frobenius reciprocity in the sense of Penney distributions.
[ 参考URL ]From a viewpoint of the orbit method, holomorphic induction is originally based on the idea of realizing an irreducible unitary representation of a Lie group $G$ in an $L^2$-space of some holomorphic sections of some line bundle over a $G$-homogeneous space associated with a polarization for a linear form of the Lie algebra of $G$. It is a generalization of ordinary induction from a unitary character; Through this process, Auslander-Kostant constructed the irreducible unitary representations of type 1, connected, simply connected solvable Lie groups.
In this talk, focusing on the class of exponential solvable Lie groups, we are concerned with holomorphically induced representations $\rho$ in some general settings.
We would like to discuss the following problems:
(1) conditions of non-vanishing of $\rho$,
(2) decomposition of $\rho$ into a direct integral of irreducible representations,
(3) Frobenius reciprocity in the sense of Penney distributions.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
2024年11月05日(火)
17:00-18:30 数理科学研究科棟(駒場) ハイブリッド開催/056号室
対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
村上 順 氏 (早稲田大学)
ダブルツイスト結び目に対応する複素化された四面体とその体積予想への応用 (JAPANESE)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
村上 順 氏 (早稲田大学)
ダブルツイスト結び目に対応する複素化された四面体とその体積予想への応用 (JAPANESE)
[ 講演概要 ]
まずダブルツイスト結び目の補空間を分割する複素化された四面体について紹介し，その後で体積予想への応用について説明する．複素化された四面体は，ダブルツイスト結び目の補空間の基本群から構成される．これはボロミアン環の補空間の半分をなす理想正８面体の変形であり，ここでの変形はボロミアン環の手術によるダブルツイスト結び目の構成に対応している．
一方で，ダブルツイスト結び目のカラードジョーンズ多項式は量子 6j 記号に少し項を加えたもので表わすことができる．Neumann-Zagier 理論により量子 6j 記号と複素化された四面体の体積とが対応することがわかるので，これをダブルツイスト結び目の体積予想の証明に応用する．そのために，カラードジョーンズ多項式の代わりに ADO 不変量を用いて，ロピタルの定理によりこれを求める．また，部分積分を用いることで，big cancellation problem と呼んでいる困難さを解消する．最後に，このケースでは鞍点法が比較的容易に適用できることを紹介する．
[ 参考URL ]まずダブルツイスト結び目の補空間を分割する複素化された四面体について紹介し，その後で体積予想への応用について説明する．複素化された四面体は，ダブルツイスト結び目の補空間の基本群から構成される．これはボロミアン環の補空間の半分をなす理想正８面体の変形であり，ここでの変形はボロミアン環の手術によるダブルツイスト結び目の構成に対応している．
一方で，ダブルツイスト結び目のカラードジョーンズ多項式は量子 6j 記号に少し項を加えたもので表わすことができる．Neumann-Zagier 理論により量子 6j 記号と複素化された四面体の体積とが対応することがわかるので，これをダブルツイスト結び目の体積予想の証明に応用する．そのために，カラードジョーンズ多項式の代わりに ADO 不変量を用いて，ロピタルの定理によりこれを求める．また，部分積分を用いることで，big cancellation problem と呼んでいる困難さを解消する．最後に，このケースでは鞍点法が比較的容易に適用できることを紹介する．
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
2024年10月29日(火)
17:00-18:30 数理科学研究科棟(駒場) ハイブリッド開催/056号室
対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
内藤 貴仁 氏 (日本工業大学)
Cartan calculus in string topology (JAPANESE)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
内藤 貴仁 氏 (日本工業大学)
Cartan calculus in string topology (JAPANESE)
[ 講演概要 ]
The homology of the free loop space of a closed oriented manifold (called the loop homology) has rich algebraic structures. In the theory of string topology due to Chas and Sullivan, it is well known that the loop homology has a structure of Gerstenhabar algebras with a multiplication called the loop product and a Lie bracket called the loop bracket. On the other hand, Kuribayashi, Wakatsuki, Yamaguchi and the speaker gave a Cartan calculus on the loop homology, which is a geometric description of a homotopy Cartan calculus in the sense of Fiorenza and Kowalzig on the Hochschild homology.
In this talk, we will investigate a relationship between the string topology operations and the Cartan calculus. Especially, we will show that the Cartan calculus can be described by using the loop product and the loop bracket with rational coefficients. As an application, the nilpotency of some loop homology classes are determined.
[ 参考URL ]The homology of the free loop space of a closed oriented manifold (called the loop homology) has rich algebraic structures. In the theory of string topology due to Chas and Sullivan, it is well known that the loop homology has a structure of Gerstenhabar algebras with a multiplication called the loop product and a Lie bracket called the loop bracket. On the other hand, Kuribayashi, Wakatsuki, Yamaguchi and the speaker gave a Cartan calculus on the loop homology, which is a geometric description of a homotopy Cartan calculus in the sense of Fiorenza and Kowalzig on the Hochschild homology.
In this talk, we will investigate a relationship between the string topology operations and the Cartan calculus. Especially, we will show that the Cartan calculus can be described by using the loop product and the loop bracket with rational coefficients. As an application, the nilpotency of some loop homology classes are determined.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
2024年10月22日(火)
17:00-18:00 オンライン開催
参加を希望される場合は、セミナーのホームページから参加登録を行って下さい。
桑垣 樹 氏 (京都大学)
On the generic existence of WKB spectral networks/Stokes graphs (JAPANESE)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
参加を希望される場合は、セミナーのホームページから参加登録を行って下さい。
桑垣 樹 氏 (京都大学)
On the generic existence of WKB spectral networks/Stokes graphs (JAPANESE)
[ 講演概要 ]
リーマン面上の二次微分の定める軌道（葉層）は、古典的な研究対象である。特に、零点を通る軌道はWKB解析、タイヒミュラー理論、場の量子論などの関係から興味を持たれてきた。WKBスペクトルネットワーク（もしくはストークスグラフ）とは、その高階微分版である。この講演では、WKBスペクトルネットワークの存在証明について説明する。時間があれば、ラグランジュ交差フレアー理論との関係についても説明する。
[ 参考URL ]リーマン面上の二次微分の定める軌道（葉層）は、古典的な研究対象である。特に、零点を通る軌道はWKB解析、タイヒミュラー理論、場の量子論などの関係から興味を持たれてきた。WKBスペクトルネットワーク（もしくはストークスグラフ）とは、その高階微分版である。この講演では、WKBスペクトルネットワークの存在証明について説明する。時間があれば、ラグランジュ交差フレアー理論との関係についても説明する。
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
2024年10月17日(木)
17:00-18:30 数理科学研究科棟(駒場) ハイブリッド開催/056号室
開催日、開催場所にご注意下さい。対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
榎園 誠 氏 (東京大学大学院数理科学研究科)
Slope inequalities for fibered complex surfaces (JAPANESE)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
開催日、開催場所にご注意下さい。対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
榎園 誠 氏 (東京大学大学院数理科学研究科)
Slope inequalities for fibered complex surfaces (JAPANESE)
[ 講演概要 ]
Slope inequalities of fibered surfaces are important in relation to the classification of algebraic surfaces and the complex structure of Lefschetz fibrations in four-dimensional topology. It is also known that many slope inequalities for semi-stable fibered surfaces can be derived from the intersection theory on the moduli space of stable curves. In this talk, after outlining the background of these studies, I will explain how various slope inequalities can be obtained for fibered surfaces that are not necessarily semi-stable by extending the discussion of the moduli space.
[ 参考URL ]Slope inequalities of fibered surfaces are important in relation to the classification of algebraic surfaces and the complex structure of Lefschetz fibrations in four-dimensional topology. It is also known that many slope inequalities for semi-stable fibered surfaces can be derived from the intersection theory on the moduli space of stable curves. In this talk, after outlining the background of these studies, I will explain how various slope inequalities can be obtained for fibered surfaces that are not necessarily semi-stable by extending the discussion of the moduli space.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
2024年10月08日(火)
17:00-18:30 数理科学研究科棟(駒場) ハイブリッド開催/123号室
開催場所にご注意下さい。対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
今野 北斗 氏 (東京大学大学院数理科学研究科)
Dehn twists on 4-manifolds (JAPANESE)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
開催場所にご注意下さい。対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
今野 北斗 氏 (東京大学大学院数理科学研究科)
Dehn twists on 4-manifolds (JAPANESE)
[ 講演概要 ]
Dehn twists on surfaces form a basic class of diffeomorphisms. On 4-manifolds, an analogue of Dehn twist can be defined by considering twists along Seifert fibered 3-manifolds. In this talk, I will explain how this type of diffeomorphism exhibits interesting properties from the perspective of differential topology, and occasionally from the viewpoint of symplectic geometry as well. The proof involves gauge theory for families. This talk includes joint work with Abhishek Mallick and Masaki Taniguchi, as well as joint work with Jianfeng Lin, Anubhav Mukherjee, and Juan Muñoz-Echániz.
[ 参考URL ]Dehn twists on surfaces form a basic class of diffeomorphisms. On 4-manifolds, an analogue of Dehn twist can be defined by considering twists along Seifert fibered 3-manifolds. In this talk, I will explain how this type of diffeomorphism exhibits interesting properties from the perspective of differential topology, and occasionally from the viewpoint of symplectic geometry as well. The proof involves gauge theory for families. This talk includes joint work with Abhishek Mallick and Masaki Taniguchi, as well as joint work with Jianfeng Lin, Anubhav Mukherjee, and Juan Muñoz-Echániz.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
2024年07月23日(火)
17:00-18:30 数理科学研究科棟(駒場) ハイブリッド開催/056号室
対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
川室 圭子 氏 (University of Iowa)
Shortest word problem in braid theory (JAPANESE)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
川室 圭子 氏 (University of Iowa)
Shortest word problem in braid theory (JAPANESE)
[ 講演概要 ]
Given a braid element in B_n, searching for a shortest braid word representative (using the band-generators) is called the Shortest Braid Problem. Up to braid index n = 4, this problem has been solved by Kang, Ko, and Lee in 1997. In this talk I will discuss recent development of this problem for braid index 5 or higher. I will also show diagrammatic computational technique of the Left Canonical Form of a given braid, that is a key to the three fundamental problems in braid theory; the Word Problem, the Conjugacy Problem and the Shortest Word Problem. This is joint work with Rebecca Sorsen and Michele Capovilla-Searle.
[ 参考URL ]Given a braid element in B_n, searching for a shortest braid word representative (using the band-generators) is called the Shortest Braid Problem. Up to braid index n = 4, this problem has been solved by Kang, Ko, and Lee in 1997. In this talk I will discuss recent development of this problem for braid index 5 or higher. I will also show diagrammatic computational technique of the Left Canonical Form of a given braid, that is a key to the three fundamental problems in braid theory; the Word Problem, the Conjugacy Problem and the Shortest Word Problem. This is joint work with Rebecca Sorsen and Michele Capovilla-Searle.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
2024年07月09日(火)
17:00-18:00 オンライン開催
参加を希望される場合は、下記URLから参加登録を行って下さい。
中村 伊南沙 氏 (佐賀大学)
Knitted surfaces in the 4-ball and their chart description (JAPANESE)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
参加を希望される場合は、下記URLから参加登録を行って下さい。
中村 伊南沙 氏 (佐賀大学)
Knitted surfaces in the 4-ball and their chart description (JAPANESE)
[ 講演概要 ]
Knits (or BMW tangles) are tangles in a cylinder generated by generators of the BMW (Birman-Murakami-Wenzl) algebras, consisting of standard generators of the braid group and their inverses, and splices of crossings called pairs of hooks. We give a new construction of surfaces in $D^2 \times B^2$, called knitted surfaces (or BMW surfaces), that are described as the trace of deformations of knits, and we give the notion of charts for knitted surfaces, that are finite graphs in $B^2$. We show that a knitted surface has a chart description. Knitted surfaces and their chart description include 2-dimensional braids and their chart description. This is joint work with Jumpei Yasuda (Osaka University).
[ 参考URL ]Knits (or BMW tangles) are tangles in a cylinder generated by generators of the BMW (Birman-Murakami-Wenzl) algebras, consisting of standard generators of the braid group and their inverses, and splices of crossings called pairs of hooks. We give a new construction of surfaces in $D^2 \times B^2$, called knitted surfaces (or BMW surfaces), that are described as the trace of deformations of knits, and we give the notion of charts for knitted surfaces, that are finite graphs in $B^2$. We show that a knitted surface has a chart description. Knitted surfaces and their chart description include 2-dimensional braids and their chart description. This is joint work with Jumpei Yasuda (Osaka University).
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
2024年07月02日(火)
17:00-18:30 数理科学研究科棟(駒場) ハイブリッド開催/117号室
開催場所にご注意下さい。対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
田中 心 氏 (東京学芸大学)
The second quandle homology group of the knot $n$-quandle (JAPANESE)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
開催場所にご注意下さい。対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
田中 心 氏 (東京学芸大学)
The second quandle homology group of the knot $n$-quandle (JAPANESE)
[ 講演概要 ]
We compute the second quandle homology group of the knot $n$-quandle for each integer $n>1$, where the knot $n$-quandle is a certain quotient of the knot quandle (of an oriented classical knot in the $3$-sphere). Although the second quandle homology group of the knot quandle can only detect the unknot, it turns out that that of its 3-quandle can detect the unknot, the trefoil and the cinqfoil. This is a joint work with Yuta Taniguchi.
[ 参考URL ]We compute the second quandle homology group of the knot $n$-quandle for each integer $n>1$, where the knot $n$-quandle is a certain quotient of the knot quandle (of an oriented classical knot in the $3$-sphere). Although the second quandle homology group of the knot quandle can only detect the unknot, it turns out that that of its 3-quandle can detect the unknot, the trefoil and the cinqfoil. This is a joint work with Yuta Taniguchi.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
2024年06月25日(火)
17:00-18:30 数理科学研究科棟(駒場) 056号室
RIKEN iTHEMS との共同開催。対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
Emmy Murphy 氏 (University of Toronto)
Liouville symmetry groups and pseudo-isotopies (ENGLISH)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
RIKEN iTHEMS との共同開催。対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
Emmy Murphy 氏 (University of Toronto)
Liouville symmetry groups and pseudo-isotopies (ENGLISH)
[ 講演概要 ]
Even though $\mathbb{C}^n$ is the most basic symplectic manifold, when $n>2$ its compactly supported symplectomorphism group remains mysterious. For instance, we do not know if it is connected. To understand it better, one can define various subgroups of the symplectomorphism group, and a number of Serre fibrations between them. This leads us to the Liouville pseudo-isotopy group of a contact manifold, important for relating (for instance) compactly supported symplectomorphisms of $\mathbb{C}^n$, and contactomorphisms of the sphere at infinity. After explaining this background, the talk will focus on a new result: that the pseudo-isotopy group is connected, under a Liouville-vs-Weinstein hypothesis.
[ 参考URL ]Even though $\mathbb{C}^n$ is the most basic symplectic manifold, when $n>2$ its compactly supported symplectomorphism group remains mysterious. For instance, we do not know if it is connected. To understand it better, one can define various subgroups of the symplectomorphism group, and a number of Serre fibrations between them. This leads us to the Liouville pseudo-isotopy group of a contact manifold, important for relating (for instance) compactly supported symplectomorphisms of $\mathbb{C}^n$, and contactomorphisms of the sphere at infinity. After explaining this background, the talk will focus on a new result: that the pseudo-isotopy group is connected, under a Liouville-vs-Weinstein hypothesis.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
2024年06月20日(木)
17:00-18:30 数理科学研究科棟(駒場) 002号室
RIKEN iTHEMS との共同開催。開催日、開催場所にご注意下さい。対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
Dominik Inauen 氏 (University of Leipzig)
Rigidity and Flexibility of Iosmetric Embeddings (ENGLISH)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
RIKEN iTHEMS との共同開催。開催日、開催場所にご注意下さい。対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
Dominik Inauen 氏 (University of Leipzig)
Rigidity and Flexibility of Iosmetric Embeddings (ENGLISH)
[ 講演概要 ]
The problem of embedding abstract Riemannian manifolds isometrically (i.e. preserving the lengths) into Euclidean space stems from the conceptually fundamental question of whether abstract Riemannian manifolds and submanifolds of Euclidean space are the same. As it turns out, such embeddings have a drastically different behaviour at low regularity (i.e. $C^1$) than at high regularity (i.e. $C^2$). For example, by the famous Nash--Kuiper theorem it is possible to find $C^1$ isometric embeddings of the standard $2$-sphere into arbitrarily small balls in $\mathbb{R}^3$, and yet, in the $C^2$ category there is (up to translation and rotation) just one isometric embedding, namely the standard inclusion. Analoguous to the Onsager conjecture in fluid dynamics, one might ask if there is a sharp regularity threshold in the Hölder scale which distinguishes these flexible and rigid behaviours. In my talk I will review some known results and argue why the Hölder exponent 1/2 can be seen as a critical exponent in the problem.
[ 参考URL ]The problem of embedding abstract Riemannian manifolds isometrically (i.e. preserving the lengths) into Euclidean space stems from the conceptually fundamental question of whether abstract Riemannian manifolds and submanifolds of Euclidean space are the same. As it turns out, such embeddings have a drastically different behaviour at low regularity (i.e. $C^1$) than at high regularity (i.e. $C^2$). For example, by the famous Nash--Kuiper theorem it is possible to find $C^1$ isometric embeddings of the standard $2$-sphere into arbitrarily small balls in $\mathbb{R}^3$, and yet, in the $C^2$ category there is (up to translation and rotation) just one isometric embedding, namely the standard inclusion. Analoguous to the Onsager conjecture in fluid dynamics, one might ask if there is a sharp regularity threshold in the Hölder scale which distinguishes these flexible and rigid behaviours. In my talk I will review some known results and argue why the Hölder exponent 1/2 can be seen as a critical exponent in the problem.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
2024年06月11日(火)
17:00-18:30 数理科学研究科棟(駒場) ハイブリッド開催/056号室
対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
河澄 響矢 氏 (東京大学大学院数理科学研究科)
Fenchel-Nielsen 座標による Weil-Petersson シンプレクティック形式の Wolpert の公式の位相的証明 (JAPANESE)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
対面参加、オンライン参加のいずれの場合もセミナーのホームページから参加登録を行って下さい。
河澄 響矢 氏 (東京大学大学院数理科学研究科)
Fenchel-Nielsen 座標による Weil-Petersson シンプレクティック形式の Wolpert の公式の位相的証明 (JAPANESE)
[ 講演概要 ]
Wolpert は Teichmüller 空間上の Weil-Petersson シンプレクティック形式を Fenchel-Nielsen 座標で顕に記述している。この座標は曲面のパンツ分解に由来する。パンツ分解から自然にえられる胞体分割を導入することで Wolpert の記述の位相的な証明がえられる。この証明では、シンプレクティック形式がパンツ分解を定義する単純閉曲線のまわりに局所化している。
[ 参考URL ]Wolpert は Teichmüller 空間上の Weil-Petersson シンプレクティック形式を Fenchel-Nielsen 座標で顕に記述している。この座標は曲面のパンツ分解に由来する。パンツ分解から自然にえられる胞体分割を導入することで Wolpert の記述の位相的な証明がえられる。この証明では、シンプレクティック形式がパンツ分解を定義する単純閉曲線のまわりに局所化している。
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
2024年06月04日(火)
17:00-18:00 オンライン開催
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石川 勝巳 氏 (京都大学数理解析研究所)
The trapezoidal conjecture for the links of braid index 3 (JAPANESE)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
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石川 勝巳 氏 (京都大学数理解析研究所)
The trapezoidal conjecture for the links of braid index 3 (JAPANESE)
[ 講演概要 ]
The trapezoidal conjecture is a classical famous conjecture posed by Fox, which states that the coefficient sequence of the Alexander polynomial of any alternating link is trapezoidal. In this talk, we show this conjecture for any alternating links of braid index 3. Although the result holds for any choice of the orientation, we shall mainly discuss the case of the closures of alternating 3-braids with parallel orientations.
[ 参考URL ]The trapezoidal conjecture is a classical famous conjecture posed by Fox, which states that the coefficient sequence of the Alexander polynomial of any alternating link is trapezoidal. In this talk, we show this conjecture for any alternating links of braid index 3. Although the result holds for any choice of the orientation, we shall mainly discuss the case of the closures of alternating 3-braids with parallel orientations.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
2024年05月28日(火)
17:00-18:00 オンライン開催
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Andreani Petrou 氏 (沖縄科学技術大学院大学)
Knot invariants and their Harer-Zagier transform (ENGLISH)
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
参加を希望される場合は、下記URLから参加登録を行って下さい。
Andreani Petrou 氏 (沖縄科学技術大学院大学)
Knot invariants and their Harer-Zagier transform (ENGLISH)
[ 講演概要 ]
The Harer-Zagier (HZ) transform is a discrete Laplace transform that can be applied to knot polynomials, mapping them into a rational function of two variables $\lambda$ and $q$. The HZ transform of the HOMFLY-PT polynomial has a simple form, as it can be written as a sum of factorised terms. For some special families of knots, it can be fully factorised and it is completely determined by a set of exponents. There is an interesting relation between such exponents and Khovanov homology. Moreover, we conjecture that there is an 1-1 correspondence with such factorisability and a relation between the HOMFLY-PT and Kauffman polynomials. Furthermore, we suggest that by fixing the variable $\lambda= q^n$ for some "magical" exponent $n$, the HZ transform of any knot can obtain a factorised form in terms of cyclotomic polynomials. Finally, the zeros of the HZ transform show an interesting behaviour, which shall be discussed.
[ 参考URL ]The Harer-Zagier (HZ) transform is a discrete Laplace transform that can be applied to knot polynomials, mapping them into a rational function of two variables $\lambda$ and $q$. The HZ transform of the HOMFLY-PT polynomial has a simple form, as it can be written as a sum of factorised terms. For some special families of knots, it can be fully factorised and it is completely determined by a set of exponents. There is an interesting relation between such exponents and Khovanov homology. Moreover, we conjecture that there is an 1-1 correspondence with such factorisability and a relation between the HOMFLY-PT and Kauffman polynomials. Furthermore, we suggest that by fixing the variable $\lambda= q^n$ for some "magical" exponent $n$, the HZ transform of any knot can obtain a factorised form in terms of cyclotomic polynomials. Finally, the zeros of the HZ transform show an interesting behaviour, which shall be discussed.
https://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index_e.html
