Tuesday Seminar on Topology (April -- July, 2011)

[Japanese] [Past Programs]

16:30 -- 18:00 Graduate School of Mathematical Sciences, The University of Tokyo
Tea: 16:00 -- 16:30 Common Room

Last updated June 27, 2011
Information :　
Toshitake Kohno
Nariya Kawazumi

April 12 -- Room 056, 16:30 -- 18:00

Susumu Hirose (Tokyo University of Science)

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

Abstract: For a closed orientable surface standardly embedded in the 4-sphere, it was known that a diffeomorphism over this surface is extendable to the 4-sphere if and only if this diffeomorphism preserves the Rokhlin quadratic form of this surafce. In this talk, we will explain an approach to the same kind of problem for closed non-orientable surfaces.

April 26 -- Room 056, 16:30 -- 18:00, joint with "Lie groups and representation theory seminar"

Taro Yoshino (The University of Tokyo)

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 $(\tilde{X}, S)$.

May 10 -- Room 056, 16:30 -- 18:00

Tetsuya Ito (The University of Tokyo)

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.

May 17 -- Room 056, 16:30 -- 18:00

Atsushi Ishii (University of Tsukuba)

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.

May 24 -- Room 056, 16:30 -- 18:00

Masahiko Yoshinaga (Kyoto University)

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

May 31 -- Room 056, 17:00 -- 18:00

Takehiko Morita (Osaka University)

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.

June 7 -- Room 056, 16:30 -- 18:00, joint with "Lie groups and representation theory seminar"

Masahiko Kanai (The University of Tokyo)

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.

June 14-- Room 056, 17:00 -- 18:00

Toshiki Mabuchi (Osaka University)

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.

June 28 -- Room 056, 16:30 -- 18:00

Masahiro Futaki (The University of Tokyo)

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.

July 5 -- Room 056, 16:30 -- 18:00

Catherine Oikonomides（The University of Tokyo, 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.

July 12 -- Room 056, 16:30 -- 18:00

Keiko Kawamuro (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.