Topology

2015年10月 -- 2016年2月

[English] [過去のプログラム]

17:00 -- 18:30 数理科学研究科棟(東京大学駒場キャンパス)
Tea: 16:30 -- 17:00 コモンルーム

Last updated January 12, 2016
世話係　
河野俊丈
河澄響矢
逆井卓也

10月6日 -- 056号室, 17:00 -- 18:30

齋藤翔 (Kavli IPMU)

Delooping theorem in K-theory

Abstract: There is an important special class of infinite dimensional vector spaces, formed by those called Tate vector spaces. Since their first appearance in Tate’s work on residues of differentials on curves, they have been playing important roles in several different contexts including the study of formal loop spaces and semi-infinite Hodge theory. They have more sophisticated linear algebraic invariant than finite dimensional vector spaces, for instance the dimension of a Tate vector spaces is not a single integer, but a torsor acted upon by the all integers, and the determinant of an automorphism is not a single invertible scalar, but a torsor acted upon by the all invertible scalars. In this talk I will show how a delooping theorem in K-theory provides a clarified perspective on this phenomenon, using the recently developed higher categorical framework of infinity-topoi.

10月20日 -- 056号室, 17:30 -- 18:30

Bruno Scárdua (Universidade Federal do Rio de Janeiro)

On the existence of stable compact leaves for transversely holomorphic foliations

Abstract: One of the most important results in the theory of foliations is the celebrated Local stability theorem of Reeb : A compact leaf of a foliation having finite holonomy group is stable, indeed, it admits a fundamental system of invariant neighborhoods where each leaf is compact with finite holonomy group. This result, together with the Global stability theorem of Reeb (for codimension one real foliations), has many important consequences and motivates several questions in the theory of foliations. In this talk we show how to prove:
A transversely holomorphic foliation on a compact complex manifold, exhibits a compact stable leaf if and only if the set of compact leaves is not a zero measure subset of the manifold.
This is a joint work with César Camacho.

10月27日 -- 056号室, 15:00 -- 16:30

Jianfeng Lin (UCLA)

The unfolded Seiberg-Witten-Floer spectrum and its applications

Abstract: Following Furuta's idea of finite dimensional approximation in the Seiberg-Witten theory, Manolescu defined the Seiberg-Witten-Floer stable homotopy type for rational homology three-spheres in 2003. In this talk, I will explain how to construct similar invariants for a general three-manifold and discuss some applications of these new invariants. This is a joint work with Tirasan Khandhawit and Hirofumi Sasahira.

10月27日 -- 056号室, 17:00 -- 18:30

Yuanyuan Bao (東京大学大学院数理科学研究科)

Heegaard Floer homology for graphs

Abstract: Ozsváth and Szabó defined the Heegaard Floer homology (HF) for a closed oriented 3-manifold. The definition was then generalized to links embedded in a 3-manifold and the manifolds with boundary (sutured and bordered manifolds). In the case of links, there is a beautiful combinatorial way to rewrite the original definition of HF, which was defined on a Heegaard diagram of the given link, by using grid diagram. For a balanced bipartite graph, we defined its Heegaard diagram and the HF for it. Around the same time, Harvey and O’Donnol defined the combinatorial HF for transverse graphs (see the definition in [arXiv:1506.04785v1]). In this talk, we compare these two methods.

11月10日 -- 056号室, 17:30 -- 18:30

五味清紀 (信州大学)

Topological T-duality for "Real" circle bundle

Abstract: Topological T-duality originates from T-duality in superstring theory, and is first studied by Bouwkneght, Evslin and Mathai. The duality basically consists of two parts: The first part is that, for any pair of a principal circle bundle with `H-flux', there is another `T-dual' pair on the same base space. The second part states that the twisted K-groups of the total spaces of principal circle bundles in duality are isomorphic under degree shift. This is the most simple topological T-duality following Bunke and Schick, and there are a number of generalizations. The generalization I will talk about is a topological T-duality for "Real" circle bundles, motivated by T-duality in type II orbifold string theory. In this duality, a variant of Z_2-equivariant K-theory appears.

11月17日 -- 056号室, 17:00 -- 18:30

片長敦子 (信州大学)

Topology of some three-dimensional singularities related to algebraic geometry

Abstract: In this talk, we deal with hypersurface isolated singularities. First, we will recall some topological results of singularities. Next, we will sketch the classification of singularities in algebraic geometry. Finally, we will focus on the three-dimensional case and discuss some results obtained so far.

11月24日 -- 056号室, 17:00 -- 18:30

佐藤正寿 (東京電機大学)

On the cohomology ring of the handlebody mapping class group of genus two

Abstract: The genus two handlebody mapping class group acts on a tree constructed by Kramer from the disk complex, and decomposes into an amalgamated product of two subgroups. We determine the integral cohomology ring of the genus two handlebody mapping class group by examining these two subgroups and the Mayer-Vietoris exact sequence. Using this result, we estimate the ranks of low dimensional homology groups of the genus three handlebody mapping class group.

12月1日 -- 056号室, 17:00 -- 18:30

奥田喬之 (東京大学大学院数理科学研究科)

Monodromies of splitting families for singular fibers

Abstract: A degeneration of Riemann surfaces is a family of complex curves over a disk allowed to have a singular fiber. A singular fiber may split into several simpler singular fibers under a deformation family of such families, which is called a splitting family for the singular fiber. We are interested in the topology of splitting families. For the topological types of degenerations of Riemann surfaces, it is known that there is a good relationship with the surface mapping classes, via topological monodromy. In this talk, we introduce the "topological monodromies of splitting families", and give a description of those of certain splitting families.

12月8日 -- 056号室, 17:00 -- 18:30

山田裕一 (電気通信大学)

レンズ空間手術と４次元多様体の Kirby calculus

Abstract: 「３次元球面内の結び目に沿うデーン手術でレンズ空間が生じるものを決定せよ」という問題は「レンズ空間手術」と呼ばれています。Berge のリスト(1990) が完全なリストと信じられており Heegaard Floer 理論によって進展はしたものの、解決には至っていません。手法が４次元多様体論に近づいています。その一方 Minimally twisted 5 chain link の例外的デーン手術が再確認されて、レンズ空間からのレンズ空間手術や２成分絡み目に視野が広がったりしています。
講演では、Berge のリストの多様さと規則性を紹介しつつ、異なる結び目から同じレンズ空間が生じる組で構成する４次元多様体（丹下基生氏（筑波大）との共同研究）についてお話しします。

12月15日 -- 056号室, 17:00 -- 18:30

Constantin Teleman (University of California, Berkeley)

The Curved Cartan Complex

Abstract: The Cartan model computes the equivariant cohomology of a smooth manifold X with differentiable action of a compact Lie group G, from the invariant polynomial functions on the Lie algebra with values in differential forms and a deformation of the de Rham differential. Before extracting invariants, the Cartan differential does not square to zero and is apparently meaningless. Unrecognised was the fact that the full complex is a curved algebra, computing the quotient by G of the algebra of differential forms on X. This generates, for example, a gauged version of string topology. Another instance of the construction, applied to deformation quantisation of symplectic manifolds, gives the BRST construction of the symplectic quotient. Finally, the theory for a X point with an additional quadratic curving computes the representation category of the compact group G, and this generalises to the loop group of G and even to real semi-simple groups.

1月12日 -- 056号室, 16:30 -- 18:30

川崎盛通 (東京大学大学院数理科学研究科)

重い部分集合と非可縮周期軌道

Abstract: ビランとポルテロヴィッチ、サラモンによる研究では、開シンプレクティック多様体Mとその部分集合X、M内の自由ホモトピー類αに対する相対的なシンプレクティック容量C_BPS(M,X,α)を定義した。 C_BPS(M,X,α)はM上のハミルトン函数がXで十分大きい値を取る場合にαを代表する周期軌道が存在するかという問題に関わって定義される。一方で、エントフとポルテロヴィッチは非交叉配置性の文脈でシンプレクティック多様体の「重い」部分集合というものを定義している。
本講演ではビラン・ポルテロヴィッチ・サラモン容量C_BPS(M,X,α)の有限性 (適当な設定下での周期軌道の存在)を重い部分集合を用いて示す方法について解説する。
これまでの研究では(自由ループの)ホモトピー類αを代表する周期軌道の検出には、αを代表する軌道のハミルトン・フレアー理論を用いるのが一般的であった。重い部分集合は(可縮軌道のハミルトン・フレアー理論の)スペクトル不変量を用いて定義される概念であるので、今回の手法では可縮軌道のハミルトン・フレアー理論を用いて非可縮軌道を検出することになる。

古川遼 (東京大学大学院数理科学研究科)

On codimension two contact embeddings in the standard spheres

Abstract: In this talk we consider codimension two contact embedding problem by using higher dimensional braids. First, we focus on embeddings of contact 3-manifolds to the standard S⁵ and give some results, for example, any contact structure on S³ can embed so that it is smoothly isotopic to the standard embedding. These are joint work with John Etnyre. Second, we consider the relative Euler number of codimension two contact submanifolds and its Seifert hypersurfaces which is a generalization of the self-linking number of transverse knots in contact 3-manifolds. We give a way to calculate the relative Euler number of certain contact submanifolds obtained by braids and as an application we give examples of embeddings of one contact manifold which are isotopic as smooth embeddings but not isotopic as contact embeddings in higher dimension.

1月19日 -- 056号室, 15:00 -- 16:00

山本光 (東京大学大学院数理科学研究科)

Ricci-mean curvature flows in gradient shrinking Ricci solitons

Abstract: A Ricci-mean curvature flow is a coupled parabolic PDE system of a mean curvature flow and a Ricci flow. In this talk, we consider a Ricci-mean curvature flow in a gradient shrinking Ricci soliton, and give a generalization of a well-known result of Huisken which states that if a mean curvature flow in a Euclidean space develops a singularity of type I, then its parabolic rescaling near the singular point converges to a self-shrinker.

2月16日 -- 056号室, 17:00 -- 18:30

Luc Menichi (University of Angers)

String Topology, Euler Class and TNCZ free loop fibrations

Abstract: Let M be a connected, closed oriented manifold. Chas and Sullivan have defined a loop product and a loop coproduct on H*(LM; F), the homology of the free loops on M with coefficients in the field F. By studying this loop coproduct, I will show that if the free loop fibration ΩM → LM → M is homologically trivial, i.e. the induced map i* : H*(LM; F) → H*(ΩM; F) is onto, then the Euler characteristic of M is divisible by the characteristic of the field F (or M is a point).