Topology

2011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 27, 2011
世話係　
河野俊丈
河澄響矢

4月12日 -- 056号室, 16:30 -- 18:00

廣瀬進 (東京理科大学理工学部数学科)

On diffeomorphisms over non-orientable surfaces embedded in the 4-sphere

Abstract: ４次元球面内に標準的に埋め込まれた向き付け可能曲面上の向きを保つ可微分同相写像が向きを保つ４次元球面上の可微分同相写像に拡張できるための必要十分条件は，その曲面に対する Rokhlin の２次形式を保つことであることが知られている．本講演では，向き付け不可能な閉曲面に対する同様の問題についての現在進行中の試みについて話す．

4月26日 -- 056号室, 16:30 -- 18:00, Lie群論・表現論セミナーと合同

吉野太郎 (東京大学大学院数理科学研究科)

Topological Blow-up

Abstract: Suppose that a Lie group $G$ acts on a manifold $M$. The quotient space $X:=G\backslash M$ is locally compact, but not Hausdorff in general. Our aim is to understand such a non-Hausdorff space $X$. The space $X$ has the crack $S$. Roughly speaking, $S$ is the causal subset of non-Hausdorffness of $X$, and especially $X\setminus S$ is Hausdorff.
We introduce the concept of `topological blow-up' as a `repair' of the crack. The `repaired' space $\tilde{X}$ is locally compact and Hausdorff space containing $X\setminus S$ as its open subset. Moreover, the original space $X$ can be recovered from the pair of $(\tilde{X}, S)$.

5月10日 -- 056号室, 16:30 -- 18:00

伊藤哲也（東京大学大学院数理科学研究科）

Isotated points in the space of group left orderings

Abstract: The set of all left orderings of a group G admits a natural topology. In general the space of left orderings is homeomorphic to the union of Cantor set and finitely many isolated points. In this talk I will give a new method to construct left orderings corresponding to isolated points, and will explain how such isolated orderings reflect the structures of groups.

5月17日 -- 056号室, 16:30 -- 18:00

石井敦（筑波大学大学院数理物質科学研究科）

Quandle colorings with non-commutative flows

Abstract: This is a joint work with Masahide Iwakiri, Yeonhee Jang and Kanako Oshiro. We introduce quandle coloring invariants and quandle cocycle invariants with non-commutative flows for knots, spatial graphs, handlebody-knots, where a handlebody-knot is a handlebody embedded in the $3$-sphere. Two handlebody-knots are equivalent if one can be transformed into the other by an isotopy of $S^3$. The quandle coloring (resp. cocycle) invariant is a ``twisted'' quandle coloring (resp. cocycle) invariant.

5月24日 -- 056号室, 16:30 -- 18:00

吉永正彦 (京都大学大学院理学研究科)

Minimal Stratifications for Line Arrangements

Abstract: The homotopy type of complements of complex hyperplane arrangements have a special property, so called minimality (Dimca-Papadima and Randell, around 2000). Since then several approaches based on (continuous, discrete) Morse theory have appeared. In this talk, we introduce the "dual" object, which we call minimal stratification for real two dimensional cases. A merit is that the minimal stratification can be explicitly described in terms of semi-algebraic sets. We also see associated presentation of the fundamental group. This talk is based on arXiv:1105.1857

5月31日 -- 056号室, 17:00 -- 18:00

盛田健彦 (大阪大学大学院理学研究科)

Measures with maximum total exponent and generic properties of C¹ expanding maps

Abstract: This is a joint work with Yusuke Tokunaga. Let M be an N dimensional compact connected smooth Riemannian manifold without boundary and let E^r(M,M) be the space of C^r expanding maps endowed with C^r topology. We show that each of the following properties for element T in E¹(M,M) is generic.
(1) T has a unique measure with maximum total exponent.
(2) Any measure with maximum total exponent for T has zero entropy.
(3) Any measure with maximum total exponent for T is fully supported.
On the contrary, we show that for r greater than or equal to 2, a generic element in E^r(M,M) has no fully supported measures with maximum total exponent.

6月7日 -- 056号室, 16:30 -- 18:00, Lie群論・表現論セミナーと合同

金井雅彦 (東京大学大学院数理科学研究科)

Rigidity of group actions via invariant geometric structures

Abstract: It is a homomorphism into a FINITE dimensional Lie group that is concerned with in the classical RIGIDITY theorems such as those of Mostow and Margulis. In the meantime, differentiable GROUP ACTIONS for which we ask rigidity problems is a homomorphism into a diffeomorphism group, which is a typical example of INFINITE dimensional Lie groups. The purpose of the present talk is exhibiting several rigidity theorems for group actions in which I have been involved for years. Although quite a few fields of mathematics, such as ergodic theory, the theory of smooth dynamical systems, representation theory and so on, have made remarkable contributions to rigidity problems, I would rather emphasis geometric aspects: I would focus on those rigidity phenomenon for group actions that are observed by showing that the actions have INVARIANT GEOMETRIC STRUCTURES.

6月14日 -- 056号室, 17:00 -- 18:00

満渕俊樹 (大阪大学大学院理学研究科)

Donaldson-Tian-Yau's Conjecture

Abstract: For polarized algebraic manifolds, the concept of K-stability introduced by Tian and Donaldson is conjecturally strongly correlated to the existence of constant scalar curvature metrics (or more generally extremal Kähler metrics) in the polarization class. This is known as Donaldson-Tian-Yau's conjecture. Recently, a remarkable progress has been made by many authors toward its solution. In this talk, I'll discuss the topic mainly with emphasis on the existence part of the conjecture.

6月28日 -- 056号室, 16:30 -- 18:00

二木昌宏（東京大学大学院数理科学研究科）

On a Sebastiani-Thom theorem for directed Fukaya categories

Abstract: The directed Fukaya category defined by Seidel is a " categorification" of the Milnor lattice of hypersurface singularities. Sebastiani-Thom showed that the Milnor lattice and its monodromy behave as tensor product for the sum of singularities. A directed Fukaya category version of this theorem was conjectured by Auroux-Katzarkov- Orlov (and checked for the Landau-Ginzburg mirror of P^1 \times P^1). In this talk I introduce the directed Fukaya category and show that a Sebastiani-Thom type splitting holds in the case that one of the potential is of complex dimension 1.

7月5日 -- 056号室, 16:30 -- 18:00

Catherine Oikonomides（東京大学大学院数理科学研究科, JSPS）

The C*-algebra of codimension one foliations which are almost without holonomy

Abstract: Foliation C*-algebras have been defined abstractly by Alain Connes, in the 1980s, as part of the theory of Noncommutative Geometry. However, very few concrete examples of foliation C*-algebras have been studied until now. In this talk, we want to explain how to compute the K-theory of the C*-algebra of codimension one foliations which are "almost without holonomy", meaning that the holonomy of all the noncompact leaves of the foliation is trivial. Such foliations have a fairly simple geometrical structure, which is well known thanks to theorems by Imanishi, Hector and others. We will give some concrete examples on 3-manifolds, in particular the 3-sphere with the Reeb foliation, and also some slighty more complicated examples.

7月12日 -- 056号室, 16:30 -- 18:00

川室圭子 (University of Iowa)

The self linking number and planar open book decomposition

Abstract: I will show a self linking number formula, in language of braids, for transverse knots in contact manifolds that admit planar open book decompositions. Our formula extends the Bennequin's for the standar contact 3-sphere.