2011年9月 -- 2012年2月
[English]
[過去のプログラム]
16:30 -- 18:00 数理科学研究科棟(東京大学駒場キャンパス)
Tea: 16:00 -- 16:30 コモンルーム
Last updated February 6, 2012
世話係
河野俊丈
河澄響矢
9月20日 -- 056号室, 16:30 -- 18:00
Clara Löh (Universität Regensburg)
Functorial semi-norms on singular homology
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.
10月4日 -- 056号室, 16:30 -- 18:00
松田 能文 (東京大学大学院数理科学研究科)
相対的双曲群の相対的擬凸部分群
Abstract:
群の相対的双曲性は語双曲性の一般化としてGromovにより導入された. 相対的
双曲群の例として, 有限体積を持つ完備双曲多様体の基本群が挙げられる. 語双
曲群の擬凸部分群の一般化として, 相対的双曲群の相対的擬凸部分群が定義され
る. Osinにより相対的双曲群の双曲的に埋め込まれた部分群が導入され, 付加的
な代数的性質を持つ相対的擬凸部分群として特徴づけられている. この講演では,
相対的擬凸部分群を紹介するとともに双曲的に埋め込まれた部分群に関する尾國新一氏, 山形紗恵子氏との最近の共同研究について触れる.
10月11日 -- 056号室, 17:00 -- 18:00
Gaël Meigniez (Univ. de Bretagne-Sud, 中央大学)
Making foliations of codimension one,
thirty years after Thurston's works
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.
10月25日 -- 056号室, 16:30 -- 18:00
Andrei Pajitnov (Univ. de Nantes, 東京大学大学院数理科学研究科)
Circle-valued Morse theory for complex hyperplane arrangements
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.
11月1日 -- 056号室, 16:30 -- 18:00
竹内 潔 (筑波大学)
Motivic Milnor fibers and Jordan normal forms of monodromies
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.
11月8日 -- 056号室, 16:30 -- 18:00
與倉 昭治 (鹿児島大学)
Fiberwise bordism groups and related topics
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.
11月15日 -- 056号室, 16:30 -- 18:00
François Laudenbach (Univ. de Nantes)
Singular codimension-one foliations
and twisted open books in dimension 3.
(joint work with G. Meigniez)
Abstract:
The allowed singularities are those of functions.
According to A. Haefliger (1958),
such structures on manifolds, called Γ1-structures,
are objects of a cohomological
theory with a classifying space BΓ1.
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∞(R) as a discrete group has the same homology
as the based loop space
Ω BΓ1+.
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.
Every Γ1-structure ξ on a 3-manifold M whose
normal bundle
embeds into the tangent bundle to M is Γ1-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 ξ .
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.
11月22日 -- 056号室, 16:30 -- 18:00
河野俊丈 (東京大学大学院数理科学研究科)
Quantum and homological representations of braid groups
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.
11月29日 -- 056号室, 17:00 -- 18:00
Athanase Papadopoulos (IRMA, Univ. de Strasbourg)
Mapping class group actions
Abstract:
I will describe and present some rigidity results on mapping
class group actions on spaces of foliations on surfaces, equipped with
various topologies.
12月13日 -- 056号室, 16:30 -- 18:00
Mircea Voineagu (IPMU, The University of Tokyo)
Remarks on filtrations of the singular homology of real varieties.
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.
12月20日 -- 056号室, 16:30 -- 18:00
三松 佳彦 (中央大学理工学部)
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
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.
1月17日 -- 056号室, 16:30 -- 18:00
佐藤 隆夫 (東京理科大学理学部)
On the Johnson cokernels of the mapping class group of a surface (joint work with Naoya Enomoto)
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.
2月21日 -- 056号室, 16:30 -- 18:00
見村 万佐人 (東京大学大学院数理科学研究科)
Property (TT)/T and homomorphism superrigidity into mapping class groups
Abstract:
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．」
本講演では，この超剛性の以下のような拡張を証明する：「高ランク格子を（算術的とは限らない）一般の環上の適切な行列群に置き換えた時でも，上の定理が成り立つ．」考える群の主な例は「普遍格子」と呼ばれる群であり，これは整係数有限生成可換多項式環上の特殊線型群（SL(3,Z[x])など）のことを指す．この定理を示すために，群の"性質(TT)/T"という，Kazhdanの性質(T)を強めた性質を導入する．
以上の2性質を紹介し，群の（ユニタリ表現で捻じれた係数の）コホモロジー・有界コホモロジーとの関係を説明したい．その上で， 本講演の定理の証明の概略を述べたい．