[English]

Tea: 16:30 -- 17:00 コモンルーム

世話係

河野 俊丈

河澄 響矢

北山 貴裕

逆井 卓也

4月3日 -- 056号室, 17:00 -- 18:30

入江 慶 (東京大学大学院数理科学研究科)

Abstract: Let $L$ be a Lagrangian submanifold in a symplectic vector space which is closed, oriented and spin. Using virtual fundamental chains of moduli spaces of nonconstant pseudo-holomorphic disks with boundaries on $L$, one can define a Maurer-Cartan element of a Lie bracket operation in string topology (the loop bracket) defined at chain level. This idea is due to Fukaya, who also pointed out its important consequences in symplectic topology. In this talk I will explain how to rigorously carry out this idea. Our argument is based on a string topology chain model previously introduced by the speaker, and theory of Kuranishi structures on moduli spaces of pseudo-holomorphic disks, which has been developed by Fukaya-Oh-Ohta-Ono.

4月10日 -- 056号室, 17:00 -- 18:30

遠藤 久顕 (東京工業大学)

Abstract: 2001年にPajitnov, Rudolph, Weberは， 古典的絡み目に対してMorse-Novikov数を定義し，その基本的な性質を研究した． この不変量は，Alexander多項式やNovikovホモロジーおよびそれらの 「ねじれ版」との関連から盛んに研究されている． 本講演では，２次元結び目に対してMorse-Novikov数を定義し，いくつかの性質を述べる． 特に，２次元結び目のモーション・ピクチャーやスピン構成法との関係について解説する． 尚，本講演の内容はAndrei Pajitnov氏（ナント大学）との共同研究にもとづいている．

4月17日 -- 056号室, 17:00 -- 18:30

Tamás Kálmán (東京工業大学)

Abstract: In joint work with Daniel Mathews, we examined complements of standard Seifert surfaces of special alternating links and used Honda's method to enumerate those tight contact structures on them whose dividing sets are isotopic to the link. The number turns out to be the leading coefficient of the Alexander polynomial. The proof is rather combinatorial in nature; for example, the Euler classes of the contact structures are identified with `hypertrees' in a certain hypergraph. Using earlier results with Hitoshi Murakami and Alexander Postnikov, this yields a connection between contact topology and the Homfly polynomial. We also found that the contact invariants of our structures form a basis for the sutured Floer homology of the manifold.

4月24日 -- 056号室, 17:00 -- 18:30

山本 卓宏 (東京学芸大学)

Abstract: Following the pioneering work of R.Thom, cobordism groups of smooth maps have been studied by some mathematicians. Especially, cobordism groups of Morse functions on closed manifolds was studied by O.Saeki and K.Ikegami, B.Kalmar. In this talk, we will introduce cobordism groups among Morse functions on compact manifolds with boundary and study the cobordism groups are trivial or not by using the theory of the universal complex of singular fibers of smooth maps.

5月8日 -- 056号室, 17:00 -- 18:30

Dan Cristofaro-Gardiner (University of California, Santa Cruz)

Abstract: The Weinstein conjecture states that any Reeb vector field on a closed manifold has at least one closed orbit. The three-dimensional case of this conjecture was proved by Taubes in 2007, and Hutchings and I later showed that in this case there are always at least 2 orbits. While examples exist with exactly two orbits, one expects that this lower bound can be significantly improved with additional assumptions. For example, a theorem of Hofer, Wysocki, and Zehnder states that a generic nondegenerate Reeb vector field associated to the "standard" contact structure on S^3 has either 2, or infinitely many, closed orbits. We prove that any nondegenerate Reeb vector field has 2 or infinitely many closed orbits as long as the associated contact structure has torsion first Chern class. This is joint work with Mike Hutchings and Dan Pomerleano.

5月15日 -- 056号室, 17:00 -- 18:30

岡 睦雄 (東京理科大学)

Abstract: Consider a real algebraic variety of real codimension 2 defined by $V:=\{g(\mathbf x,\mathbf y)=h(\mathbf x,\mathbf y)=0\}$ in $\mathbb C^n=\mathbb R^n\times \mathbb R^n$. Put $\mathbf z=\mathbf x+i\mathbf y$ and consider complex valued real analytic function $f=g+ih$. Replace the variables $x_1,y_1\dots, x_n,y_n$ using the equality $x_j=(z_j+\bar z_j)/2,\, y_j=(z_j-\bar z_j)/2i$. Then $f$ can be understood to be an analytic functions of $z_j,\bar z_j$. We call $f$ a mixed function. In this way, $V=\{ f(\mathbf z,\bar \mathbf z)=0\}$ and we can use the techniques of complex analytic functions and the singularity theory developed there. In this talk, we explain basic properties of the singularity of mixed hyper surface $V(f)$ and give several open questions.

5月22日 -- 056号室, 17:00 -- 18:30

高田 土満 (東京大学大学院数理科学研究科)

Abstract: Atiyah-Singerの指数定理は， 閉多様体上の解析的指数と位相的指数が一致することを主張する， 微分トポロジーの金字塔の一つである． 私の研究目標は，その指数理論の無限次元多様体版を与えることである． そのためには，できるだけ単純な場合から始めるのが自然であるため， 次の問題を考えることにした：円周Tのループ群LTが， 「固有かつ余コンパクトに」作用している無限次元多様体に対するLT同変指数理論を， KK理論的な観点から構築せよ． いまだにこの問題の解決には至っていないが，arXiv:1701.06055， arXiv:1709.06205 では，「関数空間」と見なせるHilbert空間を始めとする， 解析的指数理論を構築するのに不可欠な対象をいくつか構成した． 本講演では，この問題に対する現時点での結果を説明する．

5月29日 -- 056号室, 17:00 -- 18:30

佐藤 光樹 (東京大学大学院数理科学研究科)

Abstract: The nu+ equivalence is an equivalence relation on the knot concordance group. Hom proves that many concordance invariants derived from Heegaard Floer homology are invariant under nu+ equivalence. In this work, we introduce a partial order on nu+ equivalence classes, and study its algebraic and geometrical properties. As an application, we prove that any genus one knot is nu+ equivalent to one of the unknot, the trefoil and its mirror.

6月5日 -- 056号室, 17:00 -- 18:30

松井 宏樹 (千葉大学)

Abstract: For a topological dynamical system on the Cantor set, one can introduce its topological full group, which is a countable subgroup of the homeomorphism group of the Cantor set. The Higman-Thompson group V_n is regarded as the topological full group of the one-sided full shift over n symbols. Replacing the one-sided full shift with other dynamical systems, we obtain variants of the Higman-Thompson group. It is then natural to ask whether those generalized Higman-Thompson groups possess similar (or different) features. I would like to discuss isomorphism classes of these groups, finiteness properties, abelianizations, connections to C*-algebras and their K-theory, and so on.

6月12日 -- 056号室, 17:00 -- 18:30

三松 佳彦 (中央大学)

Abstract: This is a report on a joint work with Elmar VOGT(Freie Universität Berlin). For codimension 1 foliations, the process of turbulization, i.e., inserting a Reeb component along a closed transversal, is well-known, while for higher codimensional foliation, similar processes were not understood until around 2006.

In this talk, first we formulate the turbulization along a closed transversal. Then in our dimension setting, namely 2-dimensional foliations on 4-manifolds ((4,2)-foliations), a cohomological criterion is given for a given transversal to a foliation, which tells the turbulization is possible or not, relying on the Thurston's h-principle. Also we give cocrete geometric constructions of turbulizations.

The cohomological criterion for turbulization is deduced from a more general criterion for a given embedded surface to be a compact leaf or a closed transversal of some foliation, which is stated in terms of the euler classes of tangent and normal bndle of the foliation to look for. The anormalous cohomological solutions for certain cases suggested the geometric realization of turbulization, while the cohomological criterion is obtained by the h-principle.

Some other modifications are also formulated for (4,2)-foliations and their possibility are assured by the anormalous solutions mentioned above. For some of them, good geometric realizations are not yet known. So far the difficulty lies on the problem of the connected components of the space of representations of the surface groups to Diff S^1.

If the time permits, some special features on the h-principle for 2-dimensional foliations are also explained.

6月19日 -- 056号室

14:30 -- 16:00 [RIKEN iTHEMS と共催]

深谷 賢治 (サイモンズセンター, SUNY)

Abstract: アティヤ-フレアー予想は， ゲージ理論におけるフレアーホモロジーとラグランジアンフレアーホモロジーの間の関係に関するものである． その一つの困難は，ラグランジアンフレアーホモロジーを考えるシンプレクティック多様体が特異点を持つことである． 双対かつ同変ラグランジアンフレアーホモロジーを考えることで，この困難が解消し， 少なくともアティヤ-フレアー予想を数学的に厳密な予想として定式化できることを説明する．

17:00 -- 18:30

今野 北斗 (東京大学大学院数理科学研究科)

Abstract: Using gauge theory, more precisely SO(3)-Yang-Mills theory and Seiberg-Witten theory, I will construct characteristic classes of 4-manifold bundles. These characteristic classes are extensions of the SO(3)-Donaldson invariant and the Seiberg-Witten invariant to families of 4-manifolds, and can detect non-triviality of smooth 4-manifold bundles. The basic idea of the construction of these characteristic classes is to consider an infinite dimensional analogue of classical characteristic classes of manifold bundles, typified by the Mumford-Morita-Miller classes for surface bundles.

7月3日 -- 056号室, 17:00 -- 18:00

吉田 純 (東京大学大学院数理科学研究科)

Abstract: The categorical construction of Hochschild homology by Connes reveals that the symmetric structure on the tensor product of abelian groups is essential. It means that the categorical meaning of ad-hoc generalizations of Hochschild homology in less symmetric monoidal abelian categories remains unclear. In this talk, I will propose formulation of this problem in terms of group operads introduced by Zhang. Moreover, for each group operad G, G-symmetric versions of categories of operators will be discussed. The notion plays a key role in defining Hochschild homology for homotopy algebras; such as topological Hochschild homology.

7月10日 -- 056号室, 17:00 -- 18:30

Emmy Murphy (Northwestern University)

Abstract: Given a Stein manifold X, under what conditions can we ensure that X is symplectomorphic to C^n? For n>2 the condition of X being diffeomorphic to C^n does not suffice, and many counterexamples have been constructed which are detected by symplectic cohomology and the Fukaya category. One might conjecture that the diffeomorphism type together with a vanishing Fukaya category characterizes C^n. While this question is currently well of of reach, we present some new partial results. The main tools we'll discuss are arboreal singularities, constructable sheaf theory, and loose Legendrians -- and how they fit together to approach this question.

7月17日 -- 056号室, 17:00 -- 18:30

石川 昌治 (慶應義塾大学)

Abstract: A contact structure is a smooth distribution of hyperplanes on an odd-dimensional manifold that is non-integrable everywhere. In the case of dimension 3, there is a nice relationship between open book decompositions of 3-manifolds and contact structures up to contactomorphisms, called Giroux correspondence. A flow-spine is a spine of a 3-manifold admitting a flow such that it is transverse to the spine and the flow in the complement of the spine is diffeomorphic to a constant flow in an open ball. In this talk, we introduce some results in progress that give a correspondence between contact structures and positive flow-spines by regarding Reeb vector fields as flows of spines. This is a joint work with Y. Koda (Hiroshima) and H. Naoe (Tohoku).

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