## トポロジー火曜セミナー

開催情報 火曜日　17:00～18:30　数理科学研究科棟(駒場) 056号室 河野 俊丈, 河澄 響矢, 北山 貴裕, 逆井卓也 http://park.itc.u-tokyo.ac.jp/MSF/topology/TuesdaySeminar/index.html Tea: 16:30 - 17:00 コモンルーム

### 2012年10月23日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム

A geometric approach to the Johnson homomorphisms (JAPANESE)
[ 講演概要 ]
ジョンソン準同型を、完備化されたゴールドマン・トゥラエフ・リー双代数への
トレリ群の

### 2012年10月16日(火)

17:10-18:10   数理科学研究科棟(駒場) 056号室
Tea: 16:50 - 17:10 コモンルーム

Analytic torsion of log-Enriques surfaces (JAPANESE)
[ 講演概要 ]
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年10月09日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム

The growth series of pure Artin groups of dihedral type (JAPANESE)
[ 講演概要 ]
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年10月02日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム

Geometric flows and their self-similar solutions
(JAPANESE)
[ 講演概要 ]
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年09月04日(火)

17:00-18:00   数理科学研究科棟(駒場) 002号室
Tea: 16:30 - 17:00 コモンルーム
Piotr Nowak 氏 (the Institute of Mathematics, Polish Academy of Sciences)
Poincare inequalities, rigid groups and applications (ENGLISH)
[ 講演概要 ]
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年07月24日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
Greg McShane 氏 (Institut Fourier, Grenoble)
Orthospectra and identities (ENGLISH)
[ 講演概要 ]
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年07月17日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム

Contact structure of mixed links (JAPANESE)
[ 講演概要 ]
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年07月10日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
Marcus Werner 氏 (Kavli IPMU)
Topology in Gravitational Lensing (ENGLISH)
[ 講演概要 ]
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年06月19日(火)

17:10-18:10   数理科学研究科棟(駒場) 056号室
Tea: 16:50 - 17:10 コモンルーム

On the universal degenerating family of Riemann surfaces
over the D-M compactification of moduli space (JAPANESE)
[ 講演概要 ]
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年06月12日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム

Topological interpretation of the quandle cocycle invariants of links (JAPANESE)
[ 講演概要 ]
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年06月05日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム

A generalization of Dehn twists (JAPANESE)
[ 講演概要 ]
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年05月29日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム

Triple linking numbers and triple point numbers
of torus-covering $T^2$-links
(JAPANESE)
[ 講演概要 ]
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年05月22日(火)

17:10-18:10   数理科学研究科棟(駒場) 056号室
Tea: 16:50 - 17:10 コモンルーム

Gamma Integral Structure in Gromov-Witten theory (JAPANESE)
[ 講演概要 ]
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年05月08日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム

Infinite examples of non-Garside monoids having fundamental elements (JAPANESE)
[ 講演概要 ]
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年05月01日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム

Minimal models, formality and hard Lefschetz property of
solvmanifolds with local systems (JAPANESE)
[ 講演概要 ]
For a simply connected solvable Lie group G with a
cocompact discrete subgroup {\\Gamma}, we consider the space of
differential forms on the solvmanifold G/{\\Gamma} with values in certain
flat bundle so that this space has a structure of a differential graded
algebra(DGA). We construct Sullivan's minimal model of this DGA. This
result is an extension of Nomizu's theorem for ordinary coefficients in
the nilpotent case. By using this result, we refine Hasegawa's result of
formality of nilmanifolds and Benson-Gordon's result of hard Lefschetz
properties of nilmanifolds.

### 2012年04月24日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
Dylan Thurston 氏 (Columbia University)
Combinatorial Heegaard Floer homology (ENGLISH)
[ 講演概要 ]
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年04月17日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
Eriko Hironaka 氏 (Florida State University)
Pseudo-Anosov mapping classes with small dilatation (ENGLISH)
[ 講演概要 ]
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年04月10日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム

On homology of symplectic derivation Lie algebras of
the free associative algebra and the free Lie algebra (JAPANESE)
[ 講演概要 ]
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年02月21日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム

Property (TT)/T and homomorphism superrigidity into mapping class groups (JAPANESE)
[ 講演概要 ]
コンパクトで向きづけられた曲面(パンクがあってもよい)の写像類群には,多くの謎めいた性質があることが知られている:写像類群はある場合には高ランク格子(つまり,高ランク代数群の既約格子)に近いふるまいをするが,別の場合にはランク1格子に近いふるまいをする.次に述べる定理はFarb--Kaimanovich--Masur超剛性と呼ばれており,写像類群のランク1格子に近いふるまいの顕著な例である:「高ランク格子(例えばSL(3,Z)や,SL(3,R)の余コンパクト格子など)から写像類群への任意の群準同型は有限の像をもつ.」

### 2012年01月17日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム

On the Johnson cokernels of the mapping class group of a surface (joint work with Naoya Enomoto) (JAPANESE)
[ 講演概要 ]
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年12月20日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム

5次元球面上のLawson 葉層の葉向シンプレクティック構造について (JAPANESE)
[ 講演概要 ]
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年12月13日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
Mircea Voineagu 氏 (IPMU, The University of Tokyo)
Remarks on filtrations of the singular homology of real varieties. (ENGLISH)
[ 講演概要 ]
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年11月29日(火)

17:00-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:40 - 17:00 コモンルーム
Athanase Papadopoulos 氏 (IRMA, Univ. de Strasbourg)
Mapping class group actions (ENGLISH)
[ 講演概要 ]
I will describe and present some rigidity results on mapping
class group actions on spaces of foliations on surfaces, equipped with various topologies.

### 2011年11月22日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム

Quantum and homological representations of braid groups (JAPANESE)
[ 講演概要 ]
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年11月15日(火)

16:30-18:00   数理科学研究科棟(駒場) 056号室
Tea: 16:00 - 16:30 コモンルーム
Francois Laudenbach 氏 (Univ. de Nantes)
Singular codimension-one foliations
and twisted open books in dimension 3.
(joint work with G. Meigniez)
(ENGLISH)
[ 講演概要 ]
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.