## 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

### 2009年05月19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
Mark Hamilton 氏 (東京大学大学院数理科学研究科, JSPS)
Geometric quantization of integrable systems
[ 講演概要 ]
The theory of geometric quantization is one way of producing a "quantum system" from a "classical system," and has been studied a great deal over the past several decades. It also has surprising ties to representation theory. However, despite this, there still does not exist a satisfactory theory of quantization for systems with singularities.

Geometric quantization requires the choice of a polarization; when using a real polarization to quantize a regular enough manifold, a result of Sniatycki says that the quantization can be found by counting certain objects, called Bohr-Sommerfeld fibres. However, there are many types of systems to which this result does not apply. One such type is the class of completely integrable systems, which are examples coming from mechanics that have many nice properties, but which are nontheless too singular for Sniatycki's theorem to apply.

In this talk we will explore one approach to the quantization of integrable systems, and show a Sniatycki-type relationship to Bohr-Sommerfeld fibres. However, some surprising features appear, including infinite-dimensional contributions and strong dependence on the polarization.

This is joint work with Eva Miranda.

### 2009年05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

Discrete subgroups of the group of circle diffeomorphisms
[ 講演概要 ]
Typical examples of discrete subgroups of the group of circle diffeomorphisms
are Fuchsian groups.
In this talk, we construct discrete subgroups of the group of
orientation-preserving
real analytic cirlcle diffeomorphisms
which are not topologically conjugate to finite coverings of Fuchsian groups.

### 2009年04月28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 ambient metric in conformal geometry
[ 講演概要 ]
In 1985, Charles Fefferman and Robin Graham gave a method for realizing a conformal manifold of dimension n as a submanifold of a Ricci-flat Lorentz metric on a manifold of dimension n+2, which is now called the ambient space. Using this correspondence, one can construct many examples of conformal invariants and conformally invariant operators. However, if n is even, their construction of the ambient space is obstructed at the jet of order n/2 and thereby the application of the ambient space was limited. In this talk, I'll recall basic ideas of the ambient space and then explain how to avoid the difficulty and go beyond the obstruction. This is a joint work with Robin Graham.

### 2009年04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
Ivan Marin 氏 (Univ. Paris VII)
Some algebraic aspects of KZ systems
[ 講演概要 ]
Knizhnik-Zamolodchikov (KZ) systems enables one
to construct representations of (generalized)
braid groups. While this geometric construction is
now very well understood, it still brings to
attention, or helps constructing, new algebraic objects.
In this talk, we will present some of them, including an
infinitesimal version of Iwahori-Hecke algebras and a
generalization of the Krammer representations of the usual
braid groups.

### 2009年03月05日(木)

16:30-18:00   数理科学研究科棟(駒場) 056号室
いつもと曜日が異なります. Tea: 16:00 - 16:30 コモンルーム
Shicheng Wang 氏 (Peking University)
Extending surface automorphisms over 4-space
[ 講演概要 ]
Let $e: M^p\\to R^{p+2}$ be a co-dimensional 2 smooth embedding
from a closed orientable manifold to the Euclidean space and $E_e$ be the subgroup of ${\\cal M}_M$, the mapping class group
of $M$, whose elements extend over $R^{p+2}$ as self-diffeomorphisms. Then there is a spin structure
on $M$ derived from the embedding which is preserved by each $\\tau \\in E_e$.

Some applications: (1) the index $[{\\cal M}_{F_g}:E_e]\\geq 2^{2g-1}+2^{g-1}$ for any embedding $e:F_g\\to R^4$, where $F_g$
is the surface of genus $g$. (2) $[{\\cal M}_{T^p}:E_e]\\geq 2^p-1$ for any unknotted embedding
$e:T^p\\to R^{p+2}$, where $T^p$ is the $p$-dimensional torus. Those two lower bounds are known to be sharp.

This is a joint work of Ding-Liu-Wang-Yao.

### 2009年01月27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

Lagrangian Floer homology and quasi homomorphism
from the group of Hamiltonian diffeomorphism

[ 講演概要 ]
Entov-Polterovich constructed quasi homomorphism
from the group of Hamiltonian diffeomorphisms using
spectral invariant due to Oh etc.
In this talk I will explain a way to study this
quasi homomorphism by using Lagrangian Floer homology.
I will also explain its generalization to use quantum
cohomology with bulk deformation.
When applied to the case of toric manifold, it
gives an example where (infinitely) many quasi homomorphism
exists.
(Joint work with Oh-Ohta-Ono).

### 2009年01月20日(火)

16:30-17:30   数理科学研究科棟(駒場) 002号室
Tea: 16:00 - 16:30 コモンルーム

Five dimensional $K$-contact manifolds of rank 2
[ 講演概要 ]
A $K$-contact manifold is an odd dimensional manifold $M$ with a contact form $\\alpha$ whose Reeb flow preserves a Riemannian metric on $M$. For examples, the underlying manifold with the underlying contact form of a Sasakian manifold is $K$-contact. In this talk, we will state our results on classification up to surgeries, the existence of compatible Sasakian metrics and a sufficient condition to be toric for closed $5$-dimensional $K$-contact manifolds with a $T^2$ action given by the closure of the Reeb flow, which are obtained by the application of Morse theory to the contact moment map for the $T^2$ action.

### 2009年01月20日(火)

17:30-18:30   数理科学研究科棟(駒場) 002号室
Tea: 16:00 - 16:30 コモンルーム

Surface links which are coverings of a trivial torus knot (JAPANESE)
[ 講演概要 ]
We consider surface links which are in the form of coverings of a
trivial torus knot, which we will call torus-covering-links.
By definition, torus-covering-links include
spun $T^2$-knots, turned spun $T^2$-knots, and symmetry-spun tori.
We see some properties of torus-covering-links.

### 2009年01月13日(火)

16:30-17:30   数理科学研究科棟(駒場) 002号室
Tea: 16:00 - 16:30 コモンルーム

Compactification of the homeomorphism group of a graph
[ 講演概要 ]
Topological properties of homeomorphism groups, especially of finite-dimensional manifolds,
have been of interest in the area of infinite-dimensional manifold topology.
For a locally finite graph $\\Gamma$ with countably many components,
the homeomorphism group $\\mathcal{H}(\\Gamma)$
and its identity component $\\mathcal{H}_+(\\Gamma)$ are topological groups
with respect to the compact-open topology. I will define natural compactifications
$\\overline{\\mathcal{H}}(\\Gamma)$ and
$\\overline{\\mathcal{H}}_+(\\Gamma)$ of these groups and describe the
topological type of the pair $(\\overline{\\mathcal{H}}_+(\\Gamma), \\mathcal{H}_+(\\Gamma))$
using the data of $\\Gamma$. I will also discuss the topological structure of
$\\overline{\\mathcal{H}}(\\Gamma)$ where $\\Gamma$ is the circle.

### 2008年12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
Bertrand Deroin 氏 (CNRS, Orsay, Universit\'e Paris-Sud 11)
Tits alternative in $Diff^1(S^1)$
[ 講演概要 ]
The following form of Tits alternative for subgroups of
homeomorphisms of the circle has been proved by Margulis: or the group
preserve a probability measure on the circle, or it contains a free
subgroup on two generators. We will prove that if the group acts by diffeomorphisms of
class $C^1$ and does not preserve a probability measure on the circle, then
in fact it contains a subgroup topologically conjugated to a Schottky group.
This is a joint work with V. Kleptsyn and A. Navas.

### 2008年12月02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 (Lie群論・表現論セミナーと合同で開催)

Vanishing and Rigidity
[ 講演概要 ]
The aim of my talk is to reveal an unforeseen link between
the classical vanishing theorems of Matsushima and Weil, on the one hand,
andrigidity of the Weyl chamber flow, a dynamical system arising from a higher-rank
noncompact Lie group, on the other. The connection is established via
"transverse extension theorems": Roughly speaking, they claim that a tangential 1- form of the
orbit foliation of the Weyl chamber flow that is tangentially closed
(and satisfies a certain mild additional condition) can be extended to a closed 1- form on the
whole space in a canonical manner. In particular, infinitesimal
rigidity of the orbit foliation of the Weyl chamber flow is proved as an application.

### 2008年11月25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
Andrei Pajitnov 氏 (Univ. de Nantes)
Circle-valued Morse theory for knots and links
[ 講演概要 ]
We will discuss several recent developments in
this theory. In the first part of the talk we prove that the Morse-Novikov number of a knot is less than or equal to twice the tunnel number of the knot, and present consequences of this result. In the second part we report on our joint project with Hiroshi Goda on the half-transversal Morse-Novikov theory for 3-manifolds.

### 2008年11月18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

[ 講演概要 ]

ある種の力学系においては、相空間やパラメータ空間の

ある意味で剛性の問題とつながっている。この現象の説明には、
ある部分集合への再帰写像の構成から得られる、あるクラスの

くりこみ作用素の考え方が重要である。これに関して、

### 2008年11月11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
Thomas Andrew Putman 氏 (MIT)
The second rational homology group of the moduli space of curves
with level structures
[ 講演概要 ]
Let $\\Gamma$ be a finite-index subgroup of the mapping class
group of a closed genus $g$ surface that contains the Torelli group. For
instance, $\\Gamma$ can be the level $L$ subgroup or the spin mapping class
group. We show that $H_2(\\Gamma;\\Q) \\cong \\Q$ for $g \\geq 5$. A corollary
of this is that the rational Picard groups of the associated finite covers
of the moduli space of curves are equal to $\\Q$. We also prove analogous
results for surface with punctures and boundary components.

### 2008年11月04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
Misha Verbitsky 氏 (ITEP, Moscow)
Lefschetz SL(2)-action and cohomology of Kaehler manifolds
[ 講演概要 ]
Let M be compact Kaehler manifold. It is well
known that any Kaehler form generates a Lefschetz SL(2)-triple
acting on cohomology of M. This action can be used to compute
cohomology of M. If M is a hyperkaehler manifold, of real
dimension 4n, then the subalgebra of its cohomology generated by
the second cohomology is isomorphic to a polynomial algebra,
up to the middle degree.

### 2008年10月28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

Nonsmoothable group actions on spin 4-manifolds
[ 講演概要 ]
We call a locally linear group action on a topological manifold nonsmoothable
if the action is not smooth with respect to any possible smooth structure.
We show in this lecture that every closed, simply connected, spin topological 4-manifold
not homeomorphic to neither S^2\\times S^2 nor S^4 allows a nonsmoothable
group action of any cyclic group with sufficiently large prime order
which depends on the manifold.

### 2008年10月21日(火)

16:30-18:00   数理科学研究科棟(駒場) 002号室
Tea: 16:00 - 16:30 コモンルーム

On embeddings of 3-manifolds in 6-manifolds
[ 講演概要 ]
In this talk, we give a simple axiomatic definition of an invariant of
smooth embeddings of 3-manifolds in 6-manifolds.
The axiom is expressed in terms of some cobordisms of pairs of manifolds of
dimensions 6 and 3 (equipped with some cohomology class of the complement) and
the signature of 4-manifolds.
We then show that our invariant gives a unified framework for some classical
invariants in low-dimensions (Haefliger invariant, Milnor's triple
linking number, Rokhlin invariant, Casson invariant,
Takase's invariant, Skopenkov's invariants).

### 2008年10月14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
Jeffrey Herschel Giansiracusa 氏 (Oxford University)
Pontrjagin-Thom maps and the Deligne-Mumford compactification
[ 講演概要 ]
An embedding f: M -> N produces, via a construction of Pontrjagin-Thom, a map from N to the Thom space of the normal bundle over M. If f is an arbitrary map then one instead gets a map from N to the infinite loop space of the Thom spectrum of the normal bundle of f. We extend this Pontrjagin-Thom construction of wrong-way maps to differentiable stacks and use it to produce interesting maps from the Deligne-Mumford compactification of the moduli space of curves to certain infinite loop spaces. We show that these maps are surjective on mod p homology in a range of degrees. We thus produce large new families of torsion cohomology classes on the Deligne-Mumford compactification.

### 2008年07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

One-step Markov Theorem on exchange classes
[ 講演概要 ]
The main theorem in this talk claims the following.
Let L_A and L_B denote a closed a-braid and a closed b-braid,
respectively, that represent one link type.
After at most (a^2 b^2)/4 exchange moves on L_A,
we can 'see' the pair of closed braids."
In this talk, we explain the main theorem in details, and
we present some applications.
In particular, we propose a strategy to construct an algorithm
that determines whether two links are ambient isotopic.

### 2008年07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
Otto van Koert 氏 (北海道大学大学院理学研究科, JSPS)
Contact homology of left-handed stabilizations and connected sums
[ 講演概要 ]
In this talk, we will give a brief introduction to contact homology, an invariant of contact manifolds defined by counting holomorphic curves in the symplectization of a contact manifold. We shall show that certain left-handed stabilizations of contact open books have vanishing contact homology.
This is done by finding restrictions on the behavior of holomorphic curves in connected sums. An additional application of this behavior of holomorphic curves is a long exact sequence for linearized contact homology.

### 2008年07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

On the Johnson homomorphisms of the automorphism group of
a free metabelian group
[ 講演概要 ]
The main object of our research is the automorphism group of a
free group. To be brief, the Johnson homomorphisms are studied in order to
describe one by one approximations of the automorphism group of a free group
. They play important roles on the study of the homology groups of the autom
orphism group of a free group. In general, to determine their images are ver
y difficult problem. In this talk, we define the Johnson homomorphisms of th
e automorphism group of a free metabelian group, and determine their images.
Using these results, we can give a lower bound on the image of the Johnson
homomorphisms of the automorphism group of a free group.

### 2008年06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
Kenneth Shackleton 氏 (東京工業大学, JSPS)
On computing distances in the pants complex
[ 講演概要 ]
The pants complex is an accurate combinatorial
model for the Weil-Petersson metric (WP) on Teichmueller space
(Brock). One hopes that many of the geometric properties
of WP are accurately replicated in the pants complex, and
this is the source of many open questions. We compare these
in general, and then focus on the 5-holed sphere and the
2-holed torus, the first non-trivial surfaces. We arrive at
an algorithm for computing distances in the (1-skeleton of the)
pants complex of either surface.
[ 参考URL ]
http://www.is.titech.ac.jp/~kjshack5/FYEO.pdf

### 2008年06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

Multiplier ideal sheaves and Futaki invariant on toric Fano manifolds.
[ 講演概要 ]
I would like to discuss the subvarieties cut off by the multiplier
ideal sheaves (MIS) and Futaki invariant on toric Fano manifolds.
Futaki invariant is one of the necessary conditions for the existence
of Kahler-Einstein metrics on Fano manifolds,
on the other hand MIS is one of the sufficient conditions introduced by Nadel.
Especially I would like to focus on the MIS related to the Monge-Ampere equation
for Kahler-Einstein metrics on non-KE toric Fano manifolds.
The motivation of this work comes from the investigation of the
relationship with slope stability
of polarized manifolds introduced by Ross and Thomas.
This talk will be based on a part of the joint work with Akito Futaki
(arXiv:0711.0614).

### 2008年06月03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 geometry of certain slices of the character variety of a knot group
[ 講演概要 ]
joint work with Fumikazu Nagasato (Meijo University)
This talk is concerned with certain subsets in the character variety of a knot group.
These subsets are called '"slices", which are defined as a level set of a regular function associated to a meridian of a knot.
They are related to character varieties for branched covers along the knot.
Some investigations indicate that an equivariant theory for a knot is connected to a theory for branched covers via slices, for example, the equivariant signature of a knot and the equivariant Casson invariant.
In this talk, we will construct a map from slices into the character varieties for branched covers and investigate the properties.
In particular, we focus on slices called "trace-free", which are used to define the Casson-Lin invariant, and the relation to the character variety for two--fold branched cover.

### 2008年05月20日(火)

17:00-18:30   数理科学研究科棟(駒場) 056号室
Tea: 16:40 -- 17:00 コモンルーム
Jer\^ome Petit 氏 (東京工業大学, JSPS)
Turaev-Viro TQFT splitting.
[ 講演概要 ]
The Turaev-Viro invariant is a 3-manifolds invariant. It is obtained in this way :
1) we use a combinatorial description of 3-manifolds, in this case it is : triangulation / Pachner moves
2) we define a scalar thanks to a categorical data (spherical category) and a topological data (triangulation)
3)we verify that the scalar is invariant under Pachner moves, then we obtain a 3-manifolds invariant.

The Turaev-Viro invariant can also be extended to a TQFT. Roughly speaking a TQFT is a data which assigns a finite dimensional vector space to every closed surface and a linear application to every 3-manifold with boundary.

In this talk, we will give a decomposition of the Turaev-Viro TQFT. More precisely, we decompose it into blocks. These blocks are given by a group associated to the spherical category which was used to construct the Turaev-Viro invariant. We will show that these blocks define a HQFT (roughly speaking a TQFT with an "homotopical data"). This HQFT is obtained from an homotopical invariant, which is the homotopical version of the Turaev-Viro invariant. Moreover this invariant can be used to obtain the homological Turaev-Viro invariant defined by Yetter.