[English]

Tea: 16:00 -- 16:30 コモンルーム

世話係

河野俊丈

河澄響矢

4月9日 -- 056号室, 16:30 -- 18:00

藤 博之 (東京大学大学院数理科学研究科)

Abstract: 本講演では，結び目に対する色付きHOMFLYホモロジーとその漸近的振る舞いに関する研究を紹介する．近年，色付きHOMFLY多項式の圏化がスペクトル系列に基づく公理系による定義と位相的弦理論に基づく物理的定義の双方が提唱され，それらの興味深い一致が様々な形で確かめられている．我々の研究では，完全対称表現に対する色付きHOMFLYホモロジーの漸近的振る舞いに関して，体積予想と類似の解析を行い，その結果，A-多項式の一般化となる“超 A-多項式”を通じて，色付きHOMFLYホモロジーのある量子構造が見出された．本講演では，こうした圏化の側面について，物理的解釈を交えながら紹介したい．尚，本講演は S. Gukov, M. Stosic, P. Sulkowski の3氏との共同研究に基づく．

4月23日 -- 056号室, 16:30 -- 18:00

Andrei Pajitnov (Université de Nantes)

Abstract: Let X be a CW-complex, G its fundamental group, and R a repesentation of G. Any element of the first cohomology group of X gives rise to an exponential deformation of R, which can be considered as a curve in the space of representations. We show that the cohomology of X with local coefficients corresponding to the generic point of this curve is computable from a spectral sequence starting from the cohomology of X with R-twisted coefficients. We compute the differentials of the spectral sequence in terms of Massey products, and discuss some particular cases arising in Kaehler geometry when the spectral sequence degenerates. We explain the relation of these invariants and the twisted Novikov homology. This is a joint work with Toshitake Kohno.

4月30日 -- 056号室, 16:30 -- 18:00

Francis Sergeraert (L'Institut Fourier, Université de Grenoble)

Abstract: Robin Forman invented the notion of Discrete Vector Field in 1997. A recent common work with Ana Romero allowed us to discover the notion of Eilenberg-Zilber discrete vector field. Giving the topologist a totally new understanding of the fundamental tools of combinatorial algebraic topology: Eilenberg-Zilber theorem, twisted Eilenberg-Zilber theorem, Serre and Eilenberg-Moore spectral sequences, Eilenberg-MacLane correspondence between topological and algebraic classifying spaces. Gives also new efficient algorithms for Algebraic Topology, considerably improving our computer program Kenzo, devoted to Constructive Algebraic Topology. The talk is devoted to an introduction to discrete vector fields, the very simple definition of the Eilenberg-Zilber vector field, and how it can be used in various contexts.

5月7日 -- 056号室, 16:30 -- 18:00

伊藤 哲也 (京都大学数理解析研究所)

Abstract: One method to construct linear representations of braid groups is to use an action of braid groups on certain homology of local system coefficient. Many famous representations, such as Burau or Lawrence-Krammer-Bigelow representations are constructed in such a way. We show that homological intersections on such homology groups are closely related to the dual Garside structure, a remarkable combinatorial structure of braid, and prove that some representations detects the length of braids in a surprisingly simple way. This work is partially joint with Bert Wiest (Univ. Rennes1).

5月14日 -- 056号室, 16:30 -- 18:00

早野 健太 (大阪大学)

Abstract: Wrinkled fibrations on closed 4-manifolds are stable maps to closed surfaces with only indefinite singularities. Such fibrations and variants of them have been studied for the past few years to obtain new descriptions of 4-manifolds using mapping class groups. Vanishing cycles of wrinkled fibrations play a key role in these studies. In this talk, we will explain how homotopies of wrinkled fibrtions affect their vanishing cycles. Part of the results in this talk is a joint work with Stefan Behrens (Max Planck Institute for Mathematics).

5月21日 -- 056号室, 16:30 -- 18:00

Yuanyuan Bao (東京大学大学院数理科学研究科)

Abstract: A spatial graph is a smooth embedding of a graph into a given 3-manifold. We can regard a link as a particular spatial graph. So it is natural to ask whether it is possible to extend the idea of link Floer homology to define a Heegaard Floer homology for spatial graphs. In this talk, we discuss some ideas towards this question. In particular, we define a Heegaard Floer homology for bipartite spatial graphs and discuss some further observations about this construction. We remark that Harvey and O’Donnol have announced a combinatorial Floer homology for spatial graphs by considering grid diagrams.

6月4日 -- 056号室, 16:30 -- 18:00

Mustafa Korkmaz (Middle East Technical University)

Abstract: For a compact connected orientable surface, the mapping class group of it is defined as the group of isotopy classes of orientation-preserving self-diffeomorphisms of S which are identity on the boundary. The action of the mapping class group on the first homology of the surface gives rise to the classical 2g-dimensional symplectic representation. The existence of a faithful linear representation of the mapping class group is still unknown. In my talk, I will show the following three results; there is no lower dimensional (complex) linear representation, up to conjugation the symplectic representation is the unique nontrivial representation in dimension 2g, and there is no faithful linear representation of the mapping class group in dimensions up to 3g-3. I will also discuss a few applications of these theorems, including some algebraic consequences.

6月11日 -- 056号室, 16:30 -- 18:00

北山 貴裕 (東京大学大学院数理科学研究科)

Abstract: Culler and Shalen established a way to construct incompressible surfaces in a 3-manifold from ideal points of the SL_2-character variety. We present an analogous theory to construct certain kinds of branched surfaces from limit points of the SL_n-character variety. Such a branched surface induces a nontrivial presentation of the fundamental group as a 2-dimensional complex of groups. This is a joint work with Takashi Hara (Osaka University).

6月18日 -- 056号室, 16:30 -- 18:00

茂手木 公彦 (日本大学)

Abstract: A Dehn surgery is said to be left-orderable if the resulting manifold of the surgery has the left-orderable fundamental group, and a Dehn surgery is called an L-space surgery if the resulting manifold of the surgery is an L-space. We will focus on left-orderable, non-L-space surgeries on knots in the 3-sphere. Once we have a knot with left-orderable surgeries, the ``periodic construction" enables us to provide infinitely many knots with left-orderable, non-L-space surgeries. We apply the construction to present infinitely many hyperbolic knots on each of which every nontrivial surgery is a left-orderable, non-L-space surgery. This is a joint work with Masakazu Teragaito.

6月25日 -- 056号室, 17:10 -- 18:10

渡邉 忠之 (島根大学)

Abstract: In his article published in 1996, K. Fukaya constructed a 3-manifold invariant by using Morse homotopy theory. Roughly, his invariant is defined by considering several Morse functions on a 3-manifold and counting with weights the ways that the theta-graph can be immersed such that edges follow gradient lines. We generalize his construction to 3-valent graphs with arbitrary number of loops for integral homology 3-spheres. I will also discuss extension of our method to 3-manifolds with positive first Betti numbers.

7月9日 -- 056号室, 16:30 -- 18:00

Ryan Budney (University of Victoria)

Abstract: Everyone who has studied topology knows the compact 2-manifolds that embed in the 3-sphere. One dimension up, the problem of which smooth 3-manifolds embed in the 4-sphere turns out to be much more involved with a handful of partial answers. I will describe what is known at the present moment.

7月16日 -- 056号室, 17:10 -- 18:10

山田 澄生 (学習院大学)

Abstract: 本講演ではクラインおよびポアンカレ以来位相幾何学の発展に伴って多くの重要な空間を提供してきた実双曲空間の実現について、 道具立ては古典的ではあるが新しいモデルを紹介する。それらの構成法は凸幾何学と射影幾何学と密接に関連しており、数学史の観点 からも興味深いと思われる。これはAthanase Papadopoulosとの共同研究である。

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