Tuesday Seminar on Topology (April -- July, 2007)

[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 July 9, 2007
Information :　
Toshitake Kohno
Nariya Kawazumi

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

Toshiyuki Kobayashi (The University of Tokyo)

Existence Problem of Compact Locally Symmetric Spaces

Abstract: The local to global study of geometries was a major trend of 20th century geometry, with remarkable developments achieved particularly in Riemannian geometry. In contrast, in areas such as Lorentz geometry, familiar to us as the space-time of relativity theory, and more generally in pseudo-Riemannian geometry, as well as in various other kinds of geometry (symplectic, complex geometry, ...), surprising little is known about global properties of the geometry even if we impose a locally homogeneous structure.

In this talk, I will give a survey on the recent developments on the question about how the local geometric structure affects the global nature of non-Riemannian manifolds with emphasis on the existence problem of compact models of locally symmetric spaces.

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

Kiyonori Gomi (The University of Tokyo)

Realization of twisted K-theory and finite-dimensional approximation of Fredholm operators

Abstract: A problem in twisted K-theory is to realize twisted K-groups generally by means of finite-dimensional geometric objects, like vector bundles. I would like to talk about an approach toward the problem by means of Mikio Furuta's generalized vector bundles. By using a twisted version of the generalized vector bundle and a finite-dimensional approximation of Fredholm operators, I construct a group into which there exists a natural injection from the twisted K-group twisted by any third integral cohomology class.

May 8-- Room 056, 16:30 -- 18:00

Tetsuhiro Moriyama (The University of Tokyo)

On the vanishing of the Rohlin invariant

Abstract: The vanishing of the Rohlin invariant of an amphichiral integral homology $3$-sphere $M$ (i.e. $M \cong -M$) is a natural consequence of some elementary properties of the Casson invariant. In this talk, we give a new direct (and more elementary) proof of this vanishing property. The main idea comes from the definition of the degree 1 part of the Kontsevich-Kuperberg-Thurston invariant, and we progress by constructing some $7$-dimensional manifolds in which $M$ is embedded.

May 15 -- Room 056, 16:30 -- 18:00

Tadayuki Watanabe (RIMS, Kyoto University)

Kontsevich's characteristic classes for higher dimensional homology sphere bundles

Abstract: As an analogue of the perturbative Chern-Simons theory, Maxim Kontsevich constructed universal characteristic classes of smooth fiber bundles with fiber diffeomorphic to a singularly framed odd dimensional homology sphere. In this talk, I will give a sketch proof of our result on non-triviality of the Kontsevich classes for 7-dimensional homology sphere bundles.


June 5 -- Room 117, 17:00 -- 18:30

Emmanuel Giroux (ENS Lyon)

Symplectic mapping classes and fillings

Abstract: We will describe a joint work in progress with Paul Biran in which contact geometry is combined with properties of Lagrangian manifolds in subcritical Stein domains to obtain nontrivaility results for symplectic mapping classes.


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

Tian-Jun Li (University of Minnesota)

The Kodaira dimension of symplectic 4-manifolds

Abstract: Various results and questions about symplectic4-manifolds can be formulated in terms of the notion of the Kodaira dimension. In particular, we will discuss the classification and the geography problems. It is interesting to understand how it behaves under some basic constructions.Time permitting we will discuss the symplectic birational aspect of this notion and speculate how to extend it to higher dimensional manifolds.

July 3 -- Room 056, 16:30 -- 18:00

Eiko Kin (Tokyo Institute of Technology)

Two invariants of pseudo--Anosov mapping classes: hyperbolic volume vs dilatation (joint work with Mitsuhiko Takasawa)

Abstract: We concern two invariants of pseudo-Anosov mapping classes. One is the dilatation of pseudo--Anosov maps and the other is the volume of mapping tori. To study how two invariants are related, fixing a surface we represent a mapping class by using the standard generator set and compute these two for all pseudo--Anosov mapping classes with up to some word length. In the talk, we observe two properties:
(1) The ratio of the topological entropy (i.e. logarithm of the dilatation) to the volume is bounded from below by some positive constant which only depends on the surface.
(2) The conjugacy class having the minimal dilatation reaches the minimal volume.
On the observation (1), in case of the mapping class group of a once--punctured torus, we give a concrete lower bound of the ratio.

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

Danny C. Calegari (California Institute of Technology)

Combable functions, quasimorphisms, and the central limit theorem (joint with Koji Fujiwara)

Abstract: Quasimorphisms on groups are dual to stable commutator length, and detect extremal phenomena in topology and dynamics. In typical groups (even in a free group) stable commutator length is very difficult to calculate, because the space of quasimorphisms is too large to study directly without adding more structure. In this talk, we show that a large class of quasimorphisms - the so-called "counting quasimorphisms" on word-hyperbolic groups - can be effectively described using simple machines called finite state automata. From this, and from the ergodic theory of finite directed graphs, one can deduce a number of properties about the statistical distribution of the values of a counting quasimorphism on elements of the group.

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

Tomoo Matsumura (The University of Tokyo)

Orbifold Cohomology of Wreath Product Orbifolds and Cohomological HyperKahler Resolution Conjecture

Abstract: Chen-Ruan orbifold cohomology ring was introduced in 2000 as the degree zero genus zero orbifold Gromov-Witten invariants with three marked points. We will review its construction in the case of global quotient orbifolds, following Fantechi-Gottsche and Jarvis-Kaufmann-Kimura. We will describe the orbifold cohomology of wreath product orbifolds and explain its application to Ruan's cohomological hyperKahler resolution conjecture.