Researcher Profile and Settings

Master

Affiliation (Master)

• Faculty of Science　Mathematics　Mathematics

Affiliation (Master)

• Faculty of Science　Mathematics　Mathematics

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Profile and Settings

Degree

• (BLANK)（The University of Tokyo）
• (BLANK)（The University of Tokyo）

Profile and Settings

• Name (Japanese)

  Kanda

• Name (Kana)

  Yutaka

• Name

  200901090119619383

Achievement

Research Areas

• Natural sciences / Geometry

Research Experience

• 1994 - 1995 学術振興会 特別研究員

Education

•        - 1995  The University of Tokyo
•        - 1995  The University of Tokyo  Graduate School, Division of Mathematical Sciences
•        - 1993  The University of Tokyo  Faculty of Science
•        - 1993  The University of Tokyo  Faculty of Science

MISC

• The monopole equations and J-holomorphic curves on weakly convex almost Kahler 4-manifolds

  Y Kanda  TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY  353-  (6)  2215  -2243  2001  [Refereed][Not invited]
   

  We prove that a weakly convex almost Kahler 4-manifold contains a compact, non-constant J-holomorphic curve if the corresponding monopole invariant is not zero and if the corresponding line bundle is non-trivial.
• On the Thurston-Bennequin invariant of Legendrian knots and non exactness of Bennequin's inequality

  Y Kanda  INVENTIONES MATHEMATICAE  133-  (2)  227  -242  1998/08  [Refereed][Not invited]
   

  In this paper, we show that Bennequin's inequality is highly non exact with respect to Legendrian simple closed curves of certain topological types.
• The classification of tight contact structures on the 3-torus

  Kanda Yutaka  Communications in Analysis and Geometry  5-  (3)  413  -438  1997  [Refereed][Not invited]
   

  In this paper, we obtain the classification of orientable tight contact structures on the 3-torus.

Research Projects

• bounded cohomology and symplectic topology around mapping class groups

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2008 -2011 

  Author : KANDA Yutaka

   

  To investigate bounded cohomology of mapping class groups of compact surfaces and to show the boundedness of Morita-Mumford classes, I studied a certain class of finite dimensional representations of these groups, with special focus on estimating their range. Also I tried to apply Takefumi Nosaka' work, that on the interrelationship between quandle cohomology and Lefshetz fibrations over the sphere, to estimate the Gromov semi-norm of the 1^ Morita-Mumford classes.
• レフシェッツ束に関わるシンプレクティッフ・トポロジーと有界コホモロジー

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 2005 -2007 

  Author : 神田 雄高

   

  レフシェッツ束のトポロジーを写像類群の立場から研究した。具体的には、第1森田マンフォード類のグロモフセミノルムを上から評価する問題について考察した。 これに関してはシンプレクチック・トポロジーからの寄与が大きい。本研究者も最初はその線からのアプローチが有望であると考えていた。しかし結局は全く異なる方針を取る事になった。すなわち写像類群から代数群への表現を組織的に構成する方法が知られており、これを利用するのである。 写像類群の「作用する」曲面について、その有限不分岐被覆を決めるごとに、写像類群の表現が一つ定まる。各々の表現からはターゲットの代数群の適当な2-コサイクルを引き戻すことで、第1森田マンフォード類に相当する2-コサイクルが豊富に得られる。これら2-コサイクルのセミノルムを評価するときポイントとなるのは、表現の像の大きさである。だがリコリッシュの生成元たちが、各表現によってどんな元に移るかという問題が既に容易ではない。ちなみにアーベル被覆に相当する表現の場合は、像の大きさがよくわかっているが、ここからは我々の問題に関する非自明な結果はえられない。 本研究者は、表現の像がその定義から自明に分かる値域に一致するための「障害」を、代数的K理論を用いて定義した。障害の非自明性や大きさの評価は今後の課題である。
• Floor homology, singularities and deformation theory

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 2002 -2005 

  Author : ONO Kaoru, YAMAGUCHI Keizo, IZUMIYA Shyuichi, ISHIKAWA Goo, FUKAYA Kenji, OHTA Hiroshi

   

  We studied Floer theory in symplectic geometry in both theoretical foundation and applications. We extend Floer theory for Hamiltonian systems to that for Lagrangian intersections. Namely, we introduce the nation of filtered A_∞-algebras, filtered A_∞-bimodule etc. and give geometric constructions. As applications, we studied non-triviality of the Maslov class of Lagrangian embeddings, Lagrangian intersection under Hamiltonian deformations, etc. We also prove the flux conjecture using Floer theory for symplectomosphisms. This conjecture is basic for understanding how the group of Hamiltonian diffeomorphisms in the group of symplectomosphisms. These results are written up as research papers, preprints during the term of project. With Ohta, we tried to understand isolated singularities on complex surfaces through symplectic fillings of their links. In particular, we established uniquener of minimal symplectic fillings and studied its relation to Brieskorn's results for simple singularities. In the case of simple-elliptic singularities, we gave classification and interpreted Pinkham's result concerning the condition for existence of smoothings. These results are published in research journals.
• Symplectic structures and singularities

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 1999 -2001 

  Author : ONO Kaoru, KANDA Yutaka, ISHIKAWA Go-o, IZUMIYA Shyuichi, OHTA Hiroshi, FUKAYA Kenji

   

  It is not always the case that Floer homology for pairs of Lagrangian sumanifolds can be defined. We constructed the obstruction theory for defining Floer homology for pairs of Lagrangian submanifolds in order to clarify when it is defined. When all the obstruction classes vanish, Floer homology can be defined. However, it depends on a choice of so-called bounding chains. Dependence of Floer homology over bounding chains can be understood in the framework of filtered A_∞-algebra associated to Lagrangian submanifolds. This algebra controls the deformation (extended moduli) of unobstructed Lagrangian submanifolds and is important in itself. These results are presented in a preprint by Fukaya, Oh, Ohta and Ono. Ono and Ohta classified diffeomorphism types of minimal symplectic fillings of links of simple singularities and simple elliptic singularities (complex dimension 2). For an isolated singularity, the minimal resolution and the Milnor fibe, if it exists, give typical example of minimal symplectic fillings. But they a not diffeomorphic in general. In the case of simple singularity, they turn out diffeomorphic thanks to existence of the simultaneous resolution by Brieskorn. We studied this phenomenon from contact/symplectic viewpoint. Kanda also contributed in a course of this research.
• Projective contact geometry and singularity theory

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 1998 -2001 

  Author : ISHIKAWA Goo, SUWA Tatsuo, YAMAGUCHI Keizo, IZUMIYA Shyuichi, KIYOHARA Kazuyoshi, ONO Kaoru

   

  We have organized the joint works: The head investigator and the investigator Toru Morimoto (see the references); the head investigator and the investigators Reiko Miyabka and Makoto Kimura (submitted); the head Investigator and Ilya Bogaevski(see the references); the head investigator and S. Janeczko (submitted); the head investigator and V. Zakalyukin; and several other projects with investigators. For instance, we have the following result: The isotropic bifurcation problem is reduced to the classification of varieties by symplectomorphisms in the reduced space. The complete symplectic classification of Bruce-Gaffhey's plane curve singularites is provided and is applied to obtain naturally the Lagrangian openings. Moreover we study the Monge-Ampere equations from the view points of contact geometry and investigate the global structure and singularities of Monge-Ampere equations. Also we have the results on the singularities of Gauss mappings and dual varieties, appearing in the com formal geometry and Grassmannian geometry and application to differential equations. We construct developable submanifolds by using minimal submanifolds, harmonic mappings and calibrations.
• 低次元のシンプレクティック多様体とコンタクト多様体の研究

  日本学術振興会：科学研究費助成事業

  Date (from‐to) : 1999 -2000 

  Author : 神田 雄高

   

  本年度はおもに4次元シンプレクティック多様体を中心として研究した。 まず、代数曲面の分類理論における一般型の類似として、閉4次元シンプレクティック多様体の分類における一般型を、次の条件を満たすものとして定義する。 第一チャーン類の自乗が正で、その上の自己双対な調和2形式の次元が2以上である そして一般型の閉4次元シンプレクティック多様体Xに対する以下の予想を立てた。 1)Xは宮岡=ヤウの不等式をみたす。 2)あるシンプレクティツク形式Wが存在し、カノニカル直線束の第一チャーン類Kは、Wのドラーム=コホモロジー類および、Wの与える概複素構造のカノニカル直線束の第一チャーン類K'と一致する。 さらにこの問題を攻略するために、次のようなアプローチを試みた。 1)第一の予想について;カノニカル直線束の第一チャーン類Kのポアンカレ双対は、ある擬正則曲線Dによって実現される。X-Dには自然なスピン構造が入る。そこで古田によるモノポール方程式の大域的倉西写像の構成を、X-Dのエンドに柱状のリーマン計量をいれた設定のもとで行なう。正確には、Dのアルバネーゼトーラスのある部分トーラスを低空間とする大域的倉西写像の族が得られる。これに同変k理論を適用して、Xのベッチ数と符号数に関する不等式をえる。 この方法では、Dの管状近傍の境界Mのエータ不変量(正確なエータ形式)を計算する必要がある。この方法は、Mに自由に働く自然なU(1)作用によって、関数空間をモード分解し、Mをこの作用で割った時得られるリーマン面S上のディラック作用素の指数の計算に帰着させる。残念ながらこの最後の部分の計算が完了していないので、得られる不等式の形がまだ求まっていない。 2)第二の予想について;出発点はドナルドソンによるリフシッツ=ペンシル束構造の存在定理である。この与えられたリフシッツ=ペンシル束構造に対し、ほとんどの所で横断的なリフシッツ=ペンシル束構造であって、その一般ファイバーのホモロジー類がKのポアンカレ双対の正数倍になっているものが存在すれば、欲しいシンプレクティック形式の存在は比較的容易に示せる。そのようなうまいリフシッツ=ペンシル束構造の構成問題の半分、すなわちファイバー上のペンシルの連続的な族をうまく構成する事は、低空間であるリーマン面上のあるグラスマニアン・ファイバー束に対して、うまい切断をとるという問題に言い直せる。この部分は克服できたが、残りの部分、すなわち各ファイバー上のペンシルに、有理曲線によるパラメーター付けを一斉に与えて、欲しいリフシッツ=ペンシルの構造を得る部分がまだできない。
• Study of contact structures and foliations on 3-manifolds

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 1998 -1999 

  Author : MITSUHATSU Yoshihiko, MIZUTANI Tadayoshi, TAKAKURA Tatsuru, MATSUYAMA Yoshio, KANOA Yutaka, ONO Kaoru

   

  Mitsumatsu and Mizutani studied and constructed many examples of bi-contact structures with a research group of foliations. Especially, they constructed a bicontact structure on the 3-sphere which consists both of over-twisted ones. Still the realization problem of homotopy class of plane fields as such structures remains to be studied. Ono has established in a colaboration with Fukaya foundamental theory in applying the J-curves to symplectic topology, overcoming the notorious problem of negative multiples. Major consequences from this are the definition of Gromov-Witten invariants for general symplectic manifolds and the positive solution for a version of the Arnold conjecture for the same class. He and Kanda also worked on applying Seiberg-witten theory to contact topology, colaborating with Ohta, and got toplogical constraints on the symplectic filling 4-manifolds around simple singularities. This streem of works continues and is expected to make a further progress, especially in relation with the last subject of study below. Kanda studied contact structures in more toplogical way, and classified tight contact structures on 3-torus and showed nonexactness of Bennequin's inequality. Takakura and Mitsumatsu have been searching for the formalism to study contact topology by using Lagrangian/Legendrian torus, instead of looking at J-curves in the symplectization. This is based on the theory of geometric quantization, on which Takakura has been working. They found that most of major concepts in the theory of algebraic functions in one variable can be suitably translated and planted in this framework. However, studying contact topology through this remains as the next step of research to go.
• The Index Theorem and Analytic Secondary Invariants in Symplectic Geometry

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 1997 -1999 

  Author : MORIYOSHI Hitoshi, NATSUME Toshikazu, KAMETANI Yukio, MAEDA Yoshiaki, MATSUMOTO Makoto, ONO Kaoru

   

  The objective of the project are the followings: 1. Establish the elaborated Index Theorem in the framework of Noncommutative Geometry. Also study the relationship between the Index Theorem and the analytic secondary invariants like the eta invarinats and the spectral flow; 2. Apply the elaborated Index Theorem to Symplectic Geometry and study the Maslov class from the viewpoint of secondary classes. Here we state one of the results of the project, which is related to the Atiyah-Patodi-Singe Index Theorem in Noncommutative Geometry. Let X be a compact even-dimensional manifold with boundary Y. We equip X with a Riemaniann metric and assume that X is isometric to the product space Y×(-1,0) in a neighborhood of Y. We then denote by X the complete manifold obtained by attaching the half cylinder Y×[0,+∞] to X. To understand the Atiyah-Patodi-Singer Index Theorem in a framework of Noncommutative Geometry, we first introduce a notion of group quasi-action and understand X as the quotient with respect to a quasi-action of R. Next we construct a short exact sequence of CィイD1*ィエD1-algebras involved with kernel functions on X. We then define the index of operators on X as elements in a relative K-group. The short exact sequence constructed above is also interesting itself since it yields the Wiener-Hopf extension for CィイD1*ィエD1R even in the simplest case. Given the K-theoretic definition of index, we construct a relative cyclic cocycle that is related to the eta invariant of Y. This description makes clear the role of the integral on the L-polynomial and the eta invariant appeared in the Atiyah-Patodi-Singer Index Theorem, which are a priori depending on the choice of Riemannian metric on X. In short, the eta invariant appears as the transgression form connecting the local invariant with the index of an R-invariant operator on the cylinder Y×R. We also developed the research toi obtain the result that clearify the relation between the eta invarinats and the spectral flow for type II von Neumann algebras.
• Study on symplectic structures and contact structures

  Japan Society for the Promotion of Science：Grants-in-Aid for Scientific Research

  Date (from‐to) : 1997 -1998 

  Author : ONO Kaoru, TSUKADA Kazumi, OHTA Hiroshi, FUKAYA Kenji, KANDA Yutaka, IZUMIYA Shyuichi

   

  Fukaya and Ono had introduced the notion of Kuranishi structure and constructed Floer homology for periodic Hamiltonian systems and Gromov-Witten invariants. We continue this direction and proceeded to investigation on Floer homology of Lagrangian intersections and ALPHA_*-category proposed by Fukaya. In order to deal with transversality for the defining equation of the moduli space of holomorphic maps, we need to use multi-valued perturbations and should work with rational coefficients. Hence we need to give coherent orientation for various moduli spaces. We showed that a spin structure on the lagrangian submanifold gives & natural orientation on moduli spaces. We also have another problem, namely there are examples of a pair of Lagrangian submanifolds so that the usual definition of the boundary operator for Floer complex does not give the boundary operator. We consider the boundary values of holomorphic disks systematically and defined a sort of obstruction classes for defining Floer homology for pairs of Lagrangian submanifolds. It is also shown that if such obstruction classes vanish, we can modify the definition of the boundary operator and define Floer homology. This is a result due to Fukaya, Kontsevich, Oh, Ohta and Ono. Ohta and Ono studied topology of symplectically filling 4-manifold of links of simple singularities. Based on Taubes' theorem, we got a partial result in this direction. Inspired by this work, Kanda extended Taubes' result to certain non-compact symplectic manifolds.
• 接触多様体およびシンプレクティック多様体の研究
• Study of contact and symplectic manifolds