2007年10月 -- 2008年1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 January 17, 2008
世話係
河野俊丈
河澄響矢
10月9日 -- 056号室, 16:30 -- 18:00
浅岡 正幸 (京都大学大学院理学研究科)
Classification of codimension-one locally free actions of the affine group of the real line.
Abstract:
By GA, we denote the group of affine and orientation-preserving transformations
of the real line. In this talk, I will report on classification of locally free action of
GA on closed three manifolds, which I obtained recently. In 1979, E.Ghys proved
that if such an action preserves a volume, then it is smoothly conjugate to a homogeneous
action. However, it was unknown whether non-homogeneous action exists. As a
consequence of the classification, we will see that the unit tangent bundle of a closed
surface of higher genus admits a finite-parameter family of non-homogeneous actions.
10月16日 -- 056号室, 17:00 -- 18:00
二木 昭人 (東京工業大学大学院理工学研究科)
トーリック佐々木アインシュタイン多様体
概要：コンパクトなトーリック佐々木多様体に佐々木・アインシュタイ
ン計量が存在するための必要十分条件は高さ一定のトーリックダイアグラムから
Delzant 構成で作られることであることを見る．さらにその応用として，
トーリック Fano 多様体の標準直線束には完備 Ricci 平坦計量が
存在することが証明できることなどを解説する．
10月23日 -- 002号室, 16:30 -- 18:00
今井 淳 (首都大学東京)
部分球面のなす空間とその応用
概要：ｎ次元球面の中のｑ次元部分球面全体のなすグラスマン多様体は、擬リーマン構造や、
ある場合には更にシンプレクティック構造を持ち、これらは共形不変である。
これらの結び目や絡み目への幾何学的な応用について述べる．
10月30日 -- 056号室, 17:00 -- 18:00
太田 啓史 (名古屋大学大学院多元数理科学研究科)
$L_{\infty}$ action on Lagrangian filtered $A_{\infty}$ algebras.
Abstract:
I will discuss $L_{\infty}$ actions on Lagrangian filtered
$A_{\infty}$ algebras by cycles of the ambient symplectic
manifold. In the course of the construction,
I like to remark that the stable map compactification is not
sufficient in some case when we consider ones from genus zero
bordered Riemann surface. Also, if I have time, I like to discuss
some relation to (absolute) Gromov-Witten invariant and other
applications.
(This talk is based on my joint work with K.Fukaya, Y-G Oh and K. Ono.)
11月6日 -- 056号室, 16:30 -- 18:00
児玉 大樹 (東京大学大学院数理科学研究科)
サーストン不等式とオープンブック葉層構造
概要: 3次元多様体上の余次元1葉層構造を考える。葉層構造がレーブ成分を持たないとき、サーストン不等式と呼ばれる、ある種の凸性をあらわす性質をみたす[Th]。我々は、3次元多様体のオープンブック分解から定まる葉層構造に対してサーストン不等式が成り立つかどうかを検証する。
[Th] W. Thurston: Norm on the homology of 3-manifolds, Memoirs of the
AMS, 339 (1986), 99--130.
11月20日 -- 056号室, 16:30 -- 18:00
長郷 文和 (東京工業大学大学院理工学研究科)
A certain slice of the character variety of a knot group
and the knot contact homology
Abstract：
For a knot $K$ in 3-sphere, we can consider representations of
the knot group $G_K$ into $SL(2,\mathbb{C})$.
Their characters construct an algebraic set.
This is so-called the $SL(2,\mathbb{C})$-character variety of
$G_K$ and denoted by $X(G_K)$.
In this talk, we introduce a slice (a subset) $S_0(K)$ of $X(G_K)$.
In fact, this slice is closely related to the A-polynomial
and the abelian knot contact homology.
For example, the A-polynomial $A_K(m,l)$ of a knot $K$ is
a two-variable polynomial knot invariant defined by using
the character variety $X(G_K)$.
Then we can show that for any {\it small knot} $K$, the number of
irreducible components of $S_0(K)$ gives an upper bound of
the maximal degree of the A-polynomial $A_K(m,l)$ in terms of
the variable $l$.
Moreover, for any 2-bridge knot $K$, we can show that
the coordinate ring of $S_0(K)$ is exactly the degree 0
abelian knot contact homology $HC_0^{ab}(K)$.
We will mainly explain these facts.
11月27日 -- 056号室, 16:30 -- 18:00
石井 敦 (京都大学数理解析研究所)
A quandle cocycle invariant for handlebody-links
[joint work with Masahide Iwakiri (Osaka City University)]
Abstract:
A handlebody-link is a disjoint union of circles and a
finite trivalent graph embedded in a closed 3-manifold.
We consider it up to isotopies and IH-moves.
Then it represents an ambient isotopy class of
handlebodies embedded in the closed 3-manifold.
In this talk, I explain how a quandle cocycle invariant
is defined for handlebody-links.
12月4日 -- 056号室, 16:30 -- 18:00
今野 宏 (東京大学大学院数理科学研究科)
Morse theory for abelian hyperkahler quotients
概要: 1980年代、Kirwan はモース理論を用いてシンプレクティック商のベッチ数を求めた。本講演では、このアイディアをトーラスによるハイパーケーラー商の場合に拡張する。この方法では、シンプレクティック商の場合に比べ、ハイパーケーラー商の場合には、より計算が単純化される。その結果、トーラスによるハイパーケーラー商のベッチ数だけでなく、コホモロジー環も計算できることを示す。
12月11日 -- 056号室
16:30 -- 17:30
Xavier Gómez-Mont (CIMAT, Mexico)
A Singular Version of The Poincaré-Hopf Theorem
Abstract:
The Poincaré-Hopf Theorem asserts that the Euler Characteristic of a compact manifold is the sum of the indices of any vector field on it with isolated singularities.
A hypersurface in real or complex number space may be considered as the limit of the smooth hypersurfaces obtained from nearby regular values. The singularity contains “hidden” topology, which is unfolded by a smooth regeneration. At the singularity one has an algebraic invariant, the Jacobi Algebra, which is obtained by considering analytic functions modulo the partial derivatives. It contains topological information of the singularity.
One may consider vector fields tangent to a hypersurface with isolated singularities, and define topologically an index, which coincides with the sum of the Poincaré-Hopf indices of a regeneration of it tangent to a nearby smooth hypersurface.
I will explain how to compute the index of a vector field X tangent to an isolated hypersurface singularity V using Homological Algebra, as the Euler Characteristic of the homology of the complex obtained by contracting differential forms on V with the vector field X. The formula contains several terms, but the higher order terms may be translated from the invariants of the singular point to invariants in the Jacobi Algebra, making this translation a local version of the Poincaré-Hopf Theorem.
I will also explain how some of these ideas can be extended to complete intersections.
17:40 -- 18:40
Miguel A. Xicotencatl (CINVESTAV, Mexico)
Chen Ruan cohomology of cotangent orbifolds and Chas-Sullivan string topology
(Joint with: A. Gonzalez, E. Lupercio, C. Segovia, and B. Uribe)
Abstract: At the end of 90's, two theories of topology were invented roughly at the same time and attracted considerable interest in the mathematical community. One is the Chas-Sullivan's loop product on the homology of loop space and the second one is Chen-Ruan's stringy cohomology of orbifold. It was an observation of Chen that inertia orbifold (which carries Chen-Ruan cohomology) is the space of constant loops of an orbifold. Therefore, two theories should interact. In this work we show that for an interesting family of orbifolds, the virtual orbifold cohomology, turns out to be a subalgebra of the homology of the loop orbifold, and is isomorphic, as algebras, to the Chen-Ruan orbifold cohomology of its cotangent orbifold.
12月18日 -- 056号室, 16:30 -- 18:00
R.C. Penner (USC and Aarhus University)
Groupoid lifts of representations of mapping classes
Abstract: The "Ptolemy groupoid" is the fundamental path groupoid of the dual to the ideal cell decomposition of the decorated Teichmueller space of a punctured or bordered surface, and the "Torelli groupoid" is thesimilar discretization of the fundamental path groupoid of the quotient
by the Torelli subgroup of mapping classes that acts identically on the first integral homology of the surface. Mapping classes can be represented as appropriate elements of the Ptolemy groupoid and likewise for elements of the Torelli group in the Torelli groupoid.
A natural series of questions is to wonder which representations of mapping class groups, Torelli groups, and their subgroups can be lifted to the groupoid level. In a series of joint works with J. Andersen, A. Bene, N. Kawazumi, and S. Morita, we have given explicit lifts of a number of classical representations: The Johnson representations of the classical and higher Torelli groups
and the symplectic representation of the mapping class group all lift to the Torelli groupoid. Furthermore, the Nielsen representation of the mapping class group as automorphisms of a
free group lifts to the Ptolemy groupoid, and hence so too does any representation
of the mapping class group that factors through its action on the fundamental group of
the surface such as the Magnus representation. We shall survey these various groupoid lifts and discuss current and potential future applications.
1月15日 -- 056号室, 16:30 -- 17:30
飯田 修一 (東京大学大学院数理科学研究科)
Adiabatic limits of eta-invariants and the Meyer functions
Abstract: The Meyer function is the function defined on the hyperelliptic
mapping class group, which gives a signature formula for surface
bundles over surfaces.
In this talk, we introduce certain generalizations of the Meyer
function by using eta-invariants and we discuss the uniqueness of this
function and compute the values for Dehn twists.
1月29日 -- 056号室, 16:30 -- 17:30
松田 能文 (東京大学大学院数理科学研究科)
円周の微分同相のなす群の上の回転数関数
概要 : ポアンカレは、円周の向きを保つ同相写像に対して、回転数の有理性と有限軌道の存在が
同値であることを示した。この講演では、この事実が円周の向きを保つ同相写像のなすあ
る種の群に対して一般化できることを説明する。特に、円周の向きを保つ実解析的微分同
相のなす非離散的な群に対して、回転数関数による像の有限性と有限軌道の存在が同値で
あることを示す。
17:30 -- 18:30
木村 康人 (東京大学大学院数理科学研究科)
A Diagrammatic Construction of Third Homology Classes of Knot Quandles
Abstract:There exists a family of third (quandle / rack) homology classes,
called the shadow (fundamental / diagram) classes,
of the knot quandle, which are obtained from
the shadow colourings of knot diagrams.
We will show the construction of these homology classes,
and also show their relation to the shadow quandle cocycle
invariants of knots and that to other third homology classes.