2013年4月 -- 7月
[English]
[過去のプログラム]
16:30 -- 18:00 数理科学研究科棟(東京大学駒場キャンパス)
Tea: 16:00 -- 16:30 コモンルーム
Last updated June 21, 2013
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)
色付きHOMFLYホモロジーと超A-多項式
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)
Twisted Novikov homology and jump loci in formal and hyperformal spaces
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)
Discrete vector fields and fundamental algebraic topology.
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)
Homological intersection in braid group representation and dual
Garside structure
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)
Vanishing cycles and homotopies of wrinkled fibrations
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)
A Heegaard Floer homology for bipartite spatial graphs and its
properties
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)
Low-dimensional linear representations of mapping class groups.
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)
On an analogue of Culler-Shalen theory for higher-dimensional
representations
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)
Left-orderable, non-L-space surgeries on knots
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)
Higher-order generalization of Fukaya's Morse homotopy
invariant of 3-manifolds
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)
Smooth 3-manifolds in the 4-sphere
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)
実双曲空間の新しいモデルについて
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。