Tuesday Seminar on Topology

Date, time & place Tuesday 17:00 - 18:30 056Room #056 (Graduate School of Math. Sci. Bldg.) KOHNO Toshitake, KAWAZUMI Nariya, KITAYAMA Takahiro, SAKASAI Takuya Tea: 16:30 - 17:00 Common Room

Seminar information archive

2011/11/22

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Toshitake Kohno (The University of Tokyo)
Quantum and homological representations of braid groups (JAPANESE)
[ Abstract ]
Homological representations of braid groups are defined as
the action of homeomorphisms of a punctured disk on
the homology of an abelian covering of its configuration space.
These representations were extensively studied by Lawrence,
Krammer and Bigelow. In this talk we show that specializations
of the homological representations of braid groups
are equivalent to the monodromy of the KZ equation with
values in the space of null vectors in the tensor product
of Verma modules when the parameters are generic.
To prove this we use representations of the solutions of the
KZ equation by hypergeometric integrals due to Schechtman,
Varchenko and others.

In the case of special parameters these representations
are extended to quantum representations of mapping
class groups. We describe the images of such representations
and show that the images of any Johnson subgroups
contain non-abelian free groups if the genus and the
level are sufficiently large. The last part is a joint
work with Louis Funar.

2011/11/15

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Francois Laudenbach (Univ. de Nantes)
Singular codimension-one foliations
and twisted open books in dimension 3.
(joint work with G. Meigniez)
(ENGLISH)
[ Abstract ]
The allowed singularities are those of functions.
According to A. Haefliger (1958),
such structures on manifolds, called $\\Gamma_1$-structures,
are objects of a cohomological
theory with a classifying space $B\\Gamma_1$.
The problem of cancelling the singularities
(or regularization problem)
arise naturally.
For a closed manifold, it was solved by W.Thurston in a famous paper
(1976), with a proof relying on Mather's isomorphism (1971):
Diff$^\\infty(\\mathbb R)$ as a discrete group has the same homology
as the based loop space
$\\Omega B\\Gamma_1^+$.
For further extension to contact geometry, it is necessary
to solve the regularization problem
without using Mather's isomorphism.
That is what we have done in dimension 3. Our result is the following.

{\\it Every $\\Gamma_1$-structure $\\xi$ on a 3-manifold $M$ whose
normal bundle
embeds into the tangent bundle to $M$ is $\\Gamma_1$-homotopic
to a regular foliation
carried by a (possibily twisted) open book.}

The proof is elementary and relies on the dynamics of a (twisted)
pseudo-gradient of $\\xi$.

All the objects will be defined in the talk, in particular the notion
of twisted open book which is a central object in the reported paper.

2011/11/08

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Shoji Yokura (Kagoshima University )
Fiberwise bordism groups and related topics (JAPANESE)
[ Abstract ]
We have recently introduced the notion of fiberwise bordism. In this talk, after a quick review of some of the classical (co)bordism theories, we will explain motivations of considering fiberwise bordism and some results and connections with other known works, such as M. Kreck's bordism groups of orientation preserving diffeomorphisms and Emerson-Meyer's bivariant K-theory etc. An essential motivation is our recent work towards constructing a bivariant-theoretic analogue (in the sense of Fulton-MacPherson) of Levine-Morel's or Levine-Pandharipande's algebraic cobordism.

2011/11/01

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Kiyoshi Takeuchi (University of Tsukuba)
Motivic Milnor fibers and Jordan normal forms of monodromies (JAPANESE)
[ Abstract ]
We introduce a method to calculate the equivariant
Hodge-Deligne numbers of toric hypersurfaces.
Then we apply it to motivic Milnor
fibers introduced by Denef-Loeser and study the Jordan
normal forms of the local and global monodromies
of polynomials maps in various situations.
Especially we focus our attention on monodromies
at infinity studied by many people. The results will be
explicitly described by the convexity" of
the Newton polyhedra of polynomials. This is a joint work
with Y. Matsui and A. Esterov.

2011/10/25

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Andrei Pajitnov (Univ. de Nantes, Univ. of Tokyo)
Circle-valued Morse theory for complex hyperplane arrangements (ENGLISH)
[ Abstract ]
Let A be a complex hyperplane arrangement
in an n-dimensional complex vector space V.
Denote by H the union of the hyperplanes
and by M the complement to H in V.

We develop the real-valued and circle-valued Morse
theory on M. We prove that if A is essential then
M has the homotopy type of a space
obtained from a finite n-dimensional
CW complex fibered over a circle,
by attaching several cells of dimension n.

We compute the Novikov homology of M and show
that its structure is similar to the
homology with generic local coefficients:
it vanishes for all dimensions except n.

This is a joint work with Toshitake Kohno.

2011/10/11

17:00-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Gael Meigniez (Univ. de Bretagne-Sud, Chuo Univ.)
Making foliations of codimension one,
thirty years after Thurston's works
(ENGLISH)
[ Abstract ]
In 1976 Thurston proved that every closed manifold M whose
Euler characteristic is null carries a smooth foliation F of codimension
one. He actually established a h-principle allowing the regularization of
Haefliger structures through homotopy. I shall give some accounts of a new,
simpler proof of Thurston's result, not using Mather's homology equivalence; and also show that this proof allows to make F have dense leaves if dim M is at least 4. The emphasis will be put on the high dimensions.

2011/10/04

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Yoshifumi Matsuda (The University of Tokyo)
Relatively quasiconvex subgroups of relatively hyperbolic groups (JAPANESE)
[ Abstract ]
Relative hyperbolicity of groups was introduced by Gromov as a
generalization of word hyperbolicity. Motivating examples of relatively
hyperbolic groups are fundamental groups of noncompact complete
hyperbolic manifolds of finite volume. The class of relatively
quasiconvex subgroups of a realtively hyperbolic group is defined as a
genaralization of that of quasicovex subgroups of a word hyperbolic
group. The notion of hyperbolically embedded subgroups of a relatively
hyperbolic group was introduced by Osin and such groups are
characterized as relatively quasiconvex subgroups with additional
algebraic properties. In this talk I will present an introduction to
relatively quasiconvex subgroups and discuss recent joint work with Shin
-ichi Oguni and Saeko Yamagata on hyperbolically embedded subgroups.

2011/09/20

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Clara Loeh (Univ. Regensburg)
Functorial semi-norms on singular homology (ENGLISH)
[ Abstract ]
Functorial semi-norms on singular homology add metric information to
homology classes that is compatible with continuous maps. In particular,
functorial semi-norms give rise to degree theorems for certain classes
of manifolds; an invariant fitting into this context is Gromov's
simplicial volume. On the other hand, knowledge about mapping degrees
allows to construct functorial semi-norms with interesting properties;
for example, so-called inflexible simply connected manifolds give rise
to functorial semi-norms that are non-trivial on certain simply connected
spaces.

2011/07/12

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Keiko Kawamuro (University of Iowa)
The self linking number and planar open book decomposition (ENGLISH)
[ 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.

2011/07/05

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Catherine Oikonomides (The University of Tokyo, JSPS)
The C*-algebra of codimension one foliations which
are almost without holonomy (ENGLISH)
[ 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.

2011/06/28

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Masahiro Futaki (The University of Tokyo)
On a Sebastiani-Thom theorem for directed Fukaya categories (JAPANESE)
[ 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.

2011/06/14

17:00-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Toshiki Mabuchi (Osaka University)
Donaldson-Tian-Yau's Conjecture (JAPANESE)
[ 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 K\\"ahler 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.

2011/06/07

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Masahiko Kanai (The University of Tokyo)
Rigidity of group actions via invariant geometric structures (JAPANESE)
[ 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.

2011/05/31

17:00-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Takehiko Morita (Osaka University)
Measures with maximum total exponent and generic properties of $C^ {1}$ expanding maps (JAPANESE)
[ Abstract ]
This is a joint work with Yusuke Tokunaga. Let $M$ be an $N$
dimensional compact connected smooth Riemannian manifold without
boundary and let $\\mathcal{E}^{r}(M,M)$ be the space of $C^{r}$
expandig maps endowed with $C^{r}$ topology. We show that
each of the following properties for element $T$ in $\\mathcal{E} ^{1}(M,M)$ is generic.
\\begin{itemize}
\\item[(1)] $T$ has a unique measure with maximum total exponent.
\\item[(2)] Any measure with maximum total exponent for $T$ has
zero entropy.
\\item[(3)] Any measure with maximum total exponent for $T$ is
fully supported.
\\end{itemize}
On the contrary, we show that for $r\\ge 2$, a generic element
in $\\mathcal{E}^{r}(M,M)$ has no fully supported measures with
maximum total exponent.

2011/05/24

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Masahiko Yoshinaga (Kyoto University)
Minimal Stratifications for Line Arrangements (JAPANESE)
[ 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.

2011/05/17

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Atsushi Ishii (University of Tsukuba)
Quandle colorings with non-commutative flows (JAPANESE)
[ 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.

2011/05/10

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Tetsuya Ito (The University of Tokyo)
Isotated points in the space of group left orderings (JAPANESE)
[ 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.

2011/04/26

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Taro Yoshino (The University of Tokyo)
Topological Blow-up (JAPANESE)
[ 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 $(\\tilde{X}, S)$.

2011/04/12

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Susumu Hirose (Tokyo University of Science)
On diffeomorphisms over non-orientable surfaces embedded in the 4-sphere (JAPANESE)
[ 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.

2011/01/25

16:30-17:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Chikara Haruta (Graduate School of Mathematical Sciences, the University of Tokyo )
On unknotting of surface-knots with small sheet numbers
(JAPANESE)
[ Abstract ]
A connected surface smoothly embedded in ${\\mathbb R}^4$ is called a surface-knot. In particular, if a surface-knot $F$ is homeomorphic to the $2$-sphere or the torus, then it is called an $S^2$-knot or a $T^2$-knot, respectively. The sheet number of a surface-knot is an invariant analogous to the crossing number of a $1$-knot. M. Saito and S. Satoh proved some results concerning the sheet number of an $S^2$-knot. In particular, it is known that an $S^2$-knot is trivial if and only if its sheet number is $1$, and there is no $S^2$-knot whose sheet number is $2$. In this talk, we show that there is no $S^2$-knot whose sheet number is $3$, and a $T^2$-knot is trivial if and only if its sheet number is $1$.

2011/01/11

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Nariya Kawazumi (The University of Tokyo)
The Chas-Sullivan conjecture for a surface of infinite genus (JAPANESE)
[ Abstract ]
Let \\Sigma_{\\infty,1} be the inductive limit of compact
oriented surfaces with one boundary component. We prove the
center of the Goldman Lie algebra of the surface \\Sigma_{\\infty,1}
is spanned by the constant loop.
A similar statement for a closed oriented surface was conjectured
by Chas and Sullivan, and proved by Etingof. Our result is deduced
from a computation of the center of the Lie algebra of oriented chord
diagrams.
If time permits, the Lie bracket on the space of linear chord diagrams
will be discussed. This talk is based on a joint work with Yusuke Kuno
(Hiroshima U./JSPS).

2010/12/14

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Kenneth Schackleton (IPMU)
On the coarse geometry of Teichmueller space (ENGLISH)
[ Abstract ]
We discuss the synthetic geometry of the pants graph in
comparison with the Weil-Petersson metric, whose geometry the
pants graph coarsely models following work of Brock’s. We also
restrict our attention to the 5-holed sphere, studying the Gromov
bordification of the pants graph and the dynamics of pseudo-Anosov
mapping classes.

2010/12/07

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Raphael Ponge (The University of Tokyo)
Diffeomorphism-invariant geometries and noncommutative geometry (ENGLISH)
[ Abstract ]
In many geometric situations we may encounter the action of
a group $G$ on a manifold $M$, e.g., in the context of foliations. If
the action is free and proper, then the quotient $M/G$ is a smooth
manifold. However, in general the quotient $M/G$ need not even be
Hausdorff. Furthermore, it is well-known that a manifold has structure
invariant under the full group of diffeomorphisms except the
differentiable structure itself. Under these conditions how can one do
diffeomorphism-invariant geometry?

Noncommutative geometry provides us with the solution of trading the
ill-behaved space $M/G$ for a non-commutative algebra which
essentially plays the role of the algebra of smooth functions on that
space. The local index formula of Atiyah-Singer ultimately holds in
the setting of noncommutative geometry. Using this framework Connes
and Moscovici then obtained in the 90s a striking reformulation of the
local index formula in diffeomorphism-invariant geometry.

An important part the talk will be devoted to reviewing noncommutative
geometry and Connes-Moscovici's index formula. We will then hint to on-
going projects about reformulating the local index formula in two new
geometric settings: biholomorphism-invariant geometry of strictly
pseudo-convex domains and contactomorphism-invariant geometry.

2010/11/30

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Nobuhiro Nakamura (The University of Tokyo)
Pin^-(2)-monopole equations and intersection forms with local coefficients of 4-manifolds (JAPANESE)
[ Abstract ]
We introduce a variant of the Seiberg-Witten equations, Pin^-(2)-monopole equations, and explain its applications to intersection forms with local coefficients of 4-manifolds.
The first application is an analogue of Froyshov's results on 4-manifolds which have definite forms with local coefficients.
The second one is a local coefficient version of Furuta's 10/8-inequality.
As a corollary, we construct nonsmoothable spin 4-manifolds satisfying Rohlin's theorem and the 10/8-inequality.

2010/11/16

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Noboru Ito (Waseda University)
On a colored Khovanov bicomplex (JAPANESE)
[ Abstract ]
We discuss the existence of a bicomplex which is a Khovanov-type
complex associated with categorification of a colored Jones polynomial.
This is an answer to the question proposed by A. Beliakova and S. Wehrli.
Then the second term of the spectral sequence of the bicomplex corresponds
to the Khovanov-type homology group. In this talk, we explain how to define
the bicomplex. If time permits, we also define a colored Rasmussen invariant
by using another spectral sequence of the colored Jones polynomial.