2020年6月 -- 7月
[English] [過去のプログラム]
17:00 -- 18:00 Zoom でのオンライン開催
Last updated September 10, 2020
世話係
河澄 響矢
北山 貴裕
逆井 卓也
6月23日 -- Zoom でのオンライン開催, 17:00 -- 18:00
今野 北斗 (東京大学大学院数理科学研究科)
Gauge theory and the diffeomorphism and homeomorphism groups of 4-manifolds
Abstract: I will explain my recent collaboration with several groups that develops gauge theory for families
to extract difference between the diffeomorphism groups and the homeomorphism groups of 4-manifolds.
After Donaldson’s celebrated diagonalization theorem,
gauge theory has given strong constraints on the topology of smooth 4-manifolds.
Combining such constraints with Freedman’s theory, one may find many non-smoothable topological 4-manifolds.
Recently, a family version of this argument was started by T. Kato, N. Nakamura and myself,
and soon later it was developed also by D. Baraglia and his collaborating work with myself.
More precisely, considering gauge theory for smooth fiber bundles of 4-manifolds,
they obtained some constraints on the topology of smooth 4-manifold bundles.
Using such constraints, they detected non-smoothable topological fiber bundles of smooth 4-manifolds.
The existence of such bundles implies that there is homotopical difference between the diffeomorphism and homeomorphism groups of the 4-manifolds given as the fibers.
If time permits, I will also mention my collaboration with Baraglia which shows that a K3 surface gives a counterexample to the Nielsen realization problem in dimension 4.
This example reveals also that there is difference between the Nielsen realization problems asked in the smooth category and the topological category.
講演スライド
6月30日 -- Zoom でのオンライン開催, 17:00 -- 18:00
Daniel Matei (IMAR Bucharest)
Homology of right-angled Artin kernels
Abstract: The right-angled Artin groups A(G) are the finitely presented groups associated to a finite simplicial graph G=(V,E),
which are generated by the vertices V satisfying commutator relations vw=wv for every edge vw in E.
An Artin kernel Nh(G) is defined by an epimorphism h of A(G) onto the integers.
In this talk, we discuss the module structure over the Laurent polynomial ring of the homology groups of Nh(G).
7月7日 -- Zoom でのオンライン開催, 17:00 -- 18:00
野崎 雄太 (広島大学)
Abelian quotients of the Y-filtration on the homology cylinders via the LMO functor
Abstract: We construct a series of homomorphisms on the Y-filtration on the homology cylinders via the mod $＼mathbb{Z}$ reduction of the LMO functor.
The restriction of our homomorphism to the lower central series of the Torelli group
does not factor through Morita's refinement of the Johnson homomorphism.
We use it to show that the abelianization of the Johnson kernel of a closed surface has torsion elements.
This is the joint work with Masatoshi Sato and Masaaki Suzuki.
講演スライド
7月14日
[Lie群論・表現論セミナーと合同]
-- Zoom でのオンライン開催, 17:30 -- 18:30
奥田 隆幸 (広島大学)
Kobayashi's properness criterion and totally geodesic submanifolds in locally symmetric spaces
Abstract: G をリー群とし,X を G-等質空間とする.
X のいくつかの開集合を G 移動で貼り合わせて得られる多様体を(G,X)-多様体とよぶ.
X の G 不変局所幾何構造(計量など)は(G,X)-多様体に移植可能であり,
(G,X)-多様体はよい幾何構造を持った多様体の例を供給することが期待される.
この意味で, (G,X)-多様体の構成は微分幾何学における重要な研究テーマの一つである.
G の離散部分群が X に固有不連続に作用するとき,
その離散群を X の不連続群とよび, その作用による X の商多様体を Clifford--Klein 形と呼ぶ.
Clifford--Klein 形は (G,X)-多様体である.
これより G-等質空間 X 上の不連続群の構成や分類は重要な問題となる.
G-等質空間 X のイソトロピーがコンパクトである場合には,
Gの捻じれのない離散部分群はすべて不連続群である.
しかし X のイソトロピーが非コンパクトであるような場合においては,
G の捻じれのない離散群であっても, X の不連続群になるとは限らない.
以下, G が線型簡約リー群であり,
G-等質空間 X として簡約型かつイソトロピーが非コンパクトであるような場合を考える
(この設定では X は G 不変リーマンは許容しないが, G不変擬リーマン計量を許容する).
小林俊行氏は [Math.Ann.(1989)], [J. Lie Theory (1996)] において,
与えられた G の離散部分群が X の不連続群になるための判定条件を与えている.
この判定法は与えられた離散部分群と X におけるイソトロピー部分群の
``固有値の分布'' の関係性に着目する画期的なものである.
本講演では正定値非コンパクトリーマン対称空間の全測地的部分多様体の族として実現されるような G-等質空間 X について,
リーマン幾何学の言葉を用いて上記の小林氏の判定定理を翻訳したものを紹介する.
この枠組みにおいては, 与えられた離散部分群の``固有値の分布''の代わりに,
その群の定める局所対称空間の``測地ループの分布''に着目する.
7月21日 -- Zoom でのオンライン開催, 17:00 -- 18:00
Sergei Burkin (東京大学大学院数理科学研究科)
Twisted arrow categories of operads and Segal conditions
Abstract: We generalize twisted arrow category construction from categories to operads,
and show that several important categories, including the simplex category Δ,
Segal's category Γ and Moerdijk--Weiss category Ω
are twisted arrow categories of operads.
Twisted arrow categories of operads are closely connected with Segal conditions,
and the corresponding theory can be generalized from multi-object associative algebras
(i.e. categories) to multi-object P-algebras for reasonably nice operads P.
18:00 -- 19:00
Dexie Lin (東京大学大学院数理科学研究科)
Monopole Floer homology for codimension-3 Riemannian foliation
Abstract: In this paper, we give a systematic study of Seiberg-Witten theory on closed oriented manifold with codimension-3 oriented Riemannian foliation.
Under a certain topological condition, we construct the basic monopole Floer homologies for a transverse spinc structure with a bundle-like metric, generic perturbation and a complete local system.
We will show that these homologies are independent of the bundle-like metric and generic perturbation.
The major difference between the basic monopole Floer homologies and the ones on manifolds is the necessity to use the complete local system to construct the monopole Floer homologies.
7月28日 -- Zoom でのオンライン開催, 17:00 -- 18:00
Anderson Vera (京都大学数理解析研究所)
A double filtration for the mapping class group and the Goeritz group of the sphere
Abstract: I will talk about a double-indexed filtration of the mapping class group and of the Goeritz group of the sphere,
the latter is the group of isotopy classes of self-homeomorphisms of the 3-sphere which preserves the standard Heegaard splitting of S3.
In particular I will explain how this double filtration allows to write the Torelli group as a product of some subgroups of the mapping class group.
A similar study could be done for the group of automorphisms of a free group. (work in progress with K. Habiro)
講演スライド