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

2012/11/06

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Furusho Hidekazu (Nagoya University)
Galois action on knots (JAPANESE)
[ Abstract ]
I will explain a motivic structure on knots.
Then I will explain that the absolute Galois group of
the rational number field acts non-trivially
on 'the space of knots' in a non-trivial way.

2012/10/30

16:30-18:00   Room #126 (Graduate School of Math. Sci. Bldg.)
Koya Shimokawa (Saitama University)
Applications of knot theory to molecular biology (JAPANESE)
[ Abstract ]
In this talk we discuss applications of knot theory to studies of DNA
and proteins.
Especially we will consider (1)topological characterization of
mechanisms of site-specific recombination systems,
(2)modeling knotted DNA and proteins in confined regions using lattice
knots, and
(3)mechanism of topoisomerases and signed crossing changes.

2012/10/23

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Nariya Kawazumi (The University of Tokyo)
A geometric approach to the Johnson homomorphisms (JAPANESE)
[ Abstract ]
We re-construct the Johnson homomorphisms as an embeddig of the Torelli
group
into the completed Goldman-Turaev Lie bialgebra. Then the image is
included in the
kernel of the Turaev cobracket. In the case where the boundary is
connected,
the Turaev cobracket clarifies a geometric meaning of the Morita traces.
Time permitting, we also discuss the case of holed discs.
This talk is based on a joint work with Yusuke Kuno (Tsuda College).

2012/10/16

17:10-18:10   Room #056 (Graduate School of Math. Sci. Bldg.)
Ken-Ichi Yoshikawa (Kyoto University)
Analytic torsion of log-Enriques surfaces (JAPANESE)
[ Abstract ]
Log-Enriques surfaces are rational surfaces with nowhere vanishing
pluri-canonical forms. We report the recent progress on the computation
of analytic torsion of log-Enriques surfaces.

2012/10/09

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Michihiko Fujii (Kyoto University)
The growth series of pure Artin groups of dihedral type (JAPANESE)
[ Abstract ]
In this talk, I consider the kernel of the natural projection from
the Artin group of dihedral type to the corresponding Coxeter group,
that we call a pure Artin group of dihedral type,
and present rational function expressions for both the spherical and
geodesic growth series
of the pure Artin group of dihedral type with respect to a natural
generating set.
Also, I show that their growth rates are Pisot numbers.
This talk is partially based on a joint work with Takao Satoh.

2012/10/02

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Akito Futaki (The University of Tokyo)
Geometric flows and their self-similar solutions
(JAPANESE)
[ Abstract ]
In the first half of this expository talk we consider the Ricci flow and its self-similar solutions,
namely the Ricci solitons. We then specialize in the K\\"ahler case and discuss on the K\\"ahler-Einstein
problem. In the second half of this talk we consider the mean curvature flow and its self-similar
solutions, and see common aspects of the two geometric flows.

2012/09/04

17:00-18:00   Room #002 (Graduate School of Math. Sci. Bldg.)
Piotr Nowak (the Institute of Mathematics, Polish Academy of Sciences)
Poincare inequalities, rigid groups and applications (ENGLISH)
[ Abstract ]
Kazhdan’s property (T) for a group G can be expressed as a
fixed point property for affine isometric actions of G on a Hilbert
space. This definition generalizes naturally to other normed spaces. In
this talk we will focus on the spectral (aka geometric) method for
proving property (T), based on the work of Garland and studied earlier
by Pansu, Zuk, Ballmann-Swiatkowski, Dymara-Januszkiewicz
(“lambda_1>1/2” conditions) and we generalize it to to the setting of
all reflexive Banach spaces.
As applications we will show estimates of the conformal dimension of the
boundary of random hyperbolic groups in the Gromov density model and
present progress on Shalom’s conjecture on vanishing of 1-cohomology
with coefficients in uniformly bounded representations on Hilbert spaces.

2012/07/24

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Greg McShane (Institut Fourier, Grenoble)
Orthospectra and identities (ENGLISH)
[ Abstract ]
The orthospectra of a hyperbolic manifold with geodesic
boundary consists of the lengths of all geodesics perpendicular to the
boundary.
We discuss the properties of the orthospectra, asymptotics, multiplicity
and identities due to Basmajian, Bridgeman and Calegari. We will give
a proof that the identities of Bridgeman and Calegari are the same.

2012/07/17

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Mutsuo Oka (Tokyo University of Science)
Contact structure of mixed links (JAPANESE)
[ Abstract ]
A strongly non-degenerate mixed function has a Milnor open book
structures on a sufficiently small sphere. We introduce the notion of
{\\em a holomorphic-like} mixed function
and we will show that a link defined by such a mixed function has a
canonical contact structure.
Then we will show that this contact structure for a certain
holomorphic-like mixed function
is carried by the Milnor open book.

2012/07/10

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Marcus Werner (Kavli IPMU)
Topology in Gravitational Lensing (ENGLISH)
[ Abstract ]
General relativity implies that light is deflected by masses
due to the curvature of spacetime. The ensuing gravitational
lensing effect is an important tool in modern astronomy, and
topology plays a significant role in its properties. In this
talk, I will review topological aspects of gravitational lensing
theory: the connection of image numbers with Morse theory; the
interpretation of certain invariant sums of the signed image
magnification in terms of Lefschetz fixed point theory; and,
finally, a new partially topological perspective on gravitational
light deflection that emerges from the concept of optical geometry
and applications of the Gauss-Bonnet theorem.

2012/06/19

17:10-18:10   Room #056 (Graduate School of Math. Sci. Bldg.)
Yukio Matsumoto (Gakushuin University)
On the universal degenerating family of Riemann surfaces
over the D-M compactification of moduli space (JAPANESE)
[ Abstract ]
It is usually understood that over the Deligne-
Mumford compactification of moduli space of Riemann surfaces of
genus > 1, there is a family of stable curves. However, if one tries to
construct this family precisely, he/she must first take a disjoint union
of various types of smooth families of stable curves, and then divide
them by their automorphisms to paste them together. In this talk we will
show that once the smooth families are divided, the resulting quotient
family contains not only stable curves but virtually all types of
degeneration of Riemann surfaces, becoming a kind of universal
degenerating family of Riemann surfaces.

2012/06/12

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Takefumi Nosaka (RIMS, Kyoto University, JSPS)
Topological interpretation of the quandle cocycle invariants of links (JAPANESE)
[ Abstract ]
Carter et al. introduced many quandle cocycle invariants
combinatorially constructed from link-diagrams. For connected quandles of
finite order, we give a topological meaning of the invariants, without
some torsion parts. Precisely, this invariant equals a sum of "knot
colouring polynomial" and of a Z-equivariant part of the Dijkgraaf-Witten
invariant. Moreover, our approach involves applications to compute "good"
torsion subgroups of the 3-rd quandle homologies and the 2-nd homotopy
groups of rack spaces.

2012/06/05

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Yusuke Kuno (Tsuda College)
A generalization of Dehn twists (JAPANESE)
[ Abstract ]
We introduce a generalization
of Dehn twists for loops which are not
necessarily simple loops on an oriented surface.
Our generalization is an element of a certain
enlargement of the mapping class group of the surface.
A natural question is whether a generalized Dehn twist is
in the mapping class group. We show some results related to this question.
This talk is partially based on a joint work
with Nariya Kawazumi (Univ. Tokyo).

2012/05/29

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Inasa Nakamura (Gakushuin University, JSPS)
Triple linking numbers and triple point numbers
of torus-covering $T^2$-links
(JAPANESE)
[ Abstract ]
The triple linking number of an oriented surface link was defined as an
torus-covering $T^2$-link $\\mathcal{S}_m(a,b)$ is a surface link in the
form of an unbranched covering over the standard torus, determined from
two commutative $m$-braids $a$ and $b$.
In this talk, we consider $\\mathcal{S}_m(a,b)$ when $a$, $b$ are pure
$m$-braids ($m \\geq 3$), which is a surface link with $m$-components. We
present the triple linking number of $\\mathcal{S}_m(a,b)$ by using the
linking numbers of the closures of $a$ and $b$. This gives a lower bound
of the triple point number. In some cases, we can determine the triple
point numbers, each of which is a multiple of four.

2012/05/22

17:10-18:10   Room #056 (Graduate School of Math. Sci. Bldg.)
Hiroshi Iritani (Kyoto University)
Gamma Integral Structure in Gromov-Witten theory (JAPANESE)
[ Abstract ]
The quantum cohomology of a symplectic
manifold undelies a certain integral local system
defined by the Gamma characteristic class.
This local system originates from the natural integral
local sysmem on the B-side under mirror symmetry.
In this talk, I will explain its relationships to the problem
of analytic continuation of Gromov-Witten theoy (potentials),
including crepant resolution conjecture, LG/CY correspondence,
modularity in higher genus theory.

2012/05/08

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Tadashi Ishibe (The University of Tokyo, JSPS)
Infinite examples of non-Garside monoids having fundamental elements (JAPANESE)
[ Abstract ]
The Garside group, as a generalization of Artin groups,
is defined as the group of fractions of a Garside monoid.
To understand the elliptic Artin groups, which are the fundamental
groups of the complement of discriminant divisors of the semi-versal
deformation of the simply elliptic singularities E_6~, E_7~ and E_8~,
we need to consider another generalization of Artin groups.
In this talk, we will study the presentations of fundamental groups
of the complement of complexified real affine line arrangements
and consider the associated monoids.
It turns out that, in some cases, they are not Garside monoids.
Nevertheless, we will show that they satisfy the cancellation condition
and carry certain particular elements similar to the fundamental elements
in Artin monoids.
As a result, we will show that the word problem can be solved
and the center of them are determined.

2012/05/01

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Hisashi Kasuya (The University of Tokyo)
Minimal models, formality and hard Lefschetz property of
solvmanifolds with local systems (JAPANESE)

2012/04/24

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Dylan Thurston (Columbia University)
Combinatorial Heegaard Floer homology (ENGLISH)
[ Abstract ]
Heegaard Floer homology is a powerful invariant of 3- and 4-manifolds.
In 4 dimensions, Heegaard Floer homology (together with the
Seiberg-Witten and Donaldson equations, which are conjecturally
equivalent), provides essentially the only technique for
distinguishing smooth 4-manifolds. In 3 dimensions, it provides much
geometric information, like the simplest representatives of a given
homology class.

In this talk we will focus on recent progress in making Heegaard Floer
homology more computable, including a complete algorithm for computing
it for knots.

2012/04/17

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Eriko Hironaka (Florida State University)
Pseudo-Anosov mapping classes with small dilatation (ENGLISH)
[ Abstract ]
A mapping class is a homeomorphism of an oriented surface
to itself modulo isotopy. It is pseudo-Anosov if the lengths of essential
simple closed curves under iterations of the map have exponential growth
rate. The growth rate, an algebraic integer of degree bounded with
respect to the topology of the surface, is called the dilatation of the
mapping class. In this talk we will discuss the minimization problem
for dilatations of pseudo-Anosov mapping classes, and give two general
constructions of pseudo-Anosov mapping classes with small dilatation.

2012/04/10

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Takuya Sakasai (The University of Tokyo)
On homology of symplectic derivation Lie algebras of
the free associative algebra and the free Lie algebra (JAPANESE)
[ Abstract ]
We discuss homology of symplectic derivation Lie algebras of
the free associative algebra and the free Lie algebra
with particular stress on their abelianizations (degree 1 part).
Then, by using a theorem of Kontsevich,
we give some applications to rational cohomology of the moduli spaces of
Riemann surfaces and metric graphs.
This is a joint work with Shigeyuki Morita and Masaaki Suzuki.

2012/02/21

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Masato Mimura (The University of Tokyo)
Property (TT)/T and homomorphism superrigidity into mapping class groups (JAPANESE)
[ Abstract ]
Mapping class groups (MCG's), of compact oriented surfaces (possibly
with punctures), have many mysterious features: they behave not only
like higher rank lattices (namely, irreducible lattices in higher rank
algebraic groups); but also like rank one lattices. The following
theorem, the Farb--Kaimanovich--Masur superrigidity, states a rank one
phenomenon for MCG's: "every group homomorphism from higher rank
lattices (such as SL(3,Z) and cocompact lattices in SL(3,R)) into
MCG's has finite image."

In this talk, we show a generalization of the superrigidity above, to
the case where higher rank lattices are replaced with some
(non-arithmetic) matrix groups over general rings. Our main example of
such groups is called the "universal lattice", that is, the special
linear group over commutative finitely generated polynomial rings over
integers, (such as SL(3,Z[x])). To prove this, we introduce the notion
of "property (TT)/T" for groups, which is a strengthening of Kazhdan's
property (T).

We will explain these properties and relations to ordinary and bounded
cohomology of groups (with twisted unitary coefficients); and outline
the proof of our result.

2012/01/17

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Takao Satoh (Tokyo University of Science)
On the Johnson cokernels of the mapping class group of a surface (joint work with Naoya Enomoto) (JAPANESE)
[ Abstract ]
In general, the Johnson homomorphisms of the mapping class group of a surface are used to investigate graded quotients of the Johnson filtration of the mapping class group. These graded quotients are considered as a sequence of approximations of the Torelli group. Now, there is a broad range of remarkable results for the Johnson homomorphisms.
In this talk, we concentrate our focus on the cokernels of the Johnson homomorphisms of the mapping class group. By a work of Shigeyuki Morita and Hiroaki Nakamura, it is known that an Sp-irreducible module [k] appears in the cokernel of the k-th Johnson homomorphism with multiplicity one if k=2m+1 for any positive integer m. In general, however, to determine Sp-structure of the cokernel is quite a difficult preblem.
Our goal is to show that we have detected new irreducible components in the cokernels. More precisely, we will show that there appears an Sp-irreducible module [1^k] in the cokernel of the k-th Johnson homomorphism with multiplicity one if k=4m+1 for any positive integer m.

2011/12/20

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Yoshihiko Mitsumatsu (Chuo University)
Leafwise symplectic structures on Lawson's Foliation on the 5-sphere (JAPANESE)
[ Abstract ]
We are going to show that Lawson's foliation on the 5-sphere
admits a smooth leafwise symplectic sturcture. Historically, Lawson's
foliation is the first one among foliations of codimension one which are
constructed on the 5-sphere. It is obtained by modifying the Milnor
fibration associated with the Fermat type cubic polynominal in three
variables.
Alberto Verjovsky proposed a question whether if the Lawson's
foliation or slighty modified ones admit a leafwise smooth symplectic
structure and/or a leafwise complex structure. As Lawson's one has a
Kodaira-Thurston nil 4-manifold as a compact leaf, the question can not
be solved simultaneously both for the symplectic and the complex cases.
The main part of the construction is to show that the Fermat type
cubic surface admits an `end-periodic' symplectic structure, while the
natural one as an affine surface is conic at the end. Even though for
the other two families of the simple elliptic hypersurface singularities
almost the same construction works, at present, it seems very limited
where a Stein manifold admits an end-periodic symplectic structure. If
the time allows, we also discuss the existence of such structures on
globally convex symplectic manifolds.

2011/12/13

16:30-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Mircea Voineagu (IPMU, The University of Tokyo)
Remarks on filtrations of the singular homology of real varieties. (ENGLISH)
[ Abstract ]
We discuss various conjectures about filtrations on the singular homology of real and complex varieties. We prove that a conjecture relating niveau filtration on Borel-Moore homology of real varieties and the image of generalized cycle maps from reduced Lawson homology is false. In the end, we discuss a certain decomposition of Borel-Haeflinger cycle map. This is a joint work with J.Heller.

2011/11/29

17:00-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Athanase Papadopoulos (IRMA, Univ. de Strasbourg)
Mapping class group actions (ENGLISH)
[ Abstract ]
I will describe and present some rigidity results on mapping
class group actions on spaces of foliations on surfaces, equipped with various topologies.