Authorship：Coinvestigator(s) 

The aim is to study the non-commutative objects in Number theory, algebraic geometry, differential geometry, topology and mathematical physics including particle physics and to propose new idea in geometry. The features of this study is to hire the theory of deformation quantization and non-commutative geometry and develop them to apply several research topics. In particluar, we construct a new method of non-formal deformation quantization to cooperate with theoretical physics, including particle physics and string theory. We also establish an international research network in this research area.

Authorship：Collaborating Investigator(s) (not designated on Grant-in-Aid) 

Foliations and contact structures are studied, with a focus on those structures on 3, 4, and 5 dimensional spaces. Especially the construction of important examples and their mutual relationships are investigated. Interactions of many mathematical theories such as Milnor fibrations associated with singularities, fluid mechanics, symplectic geometry which is a refinement of classical mechanics, and several complex variables, re flected on those structures and objects are studied.
This study is also understood as looking at the tightness of those structures which is interpreted as the result of such mathematical theories are reflected on the geodesic flows of surfaces, which gives rise to a special class of flows called `Anosov flows'.

Authorship：Principal investigator 

Grant amount：\4160000 （ Direct Cost: \3200000 、 Indirect Cost：\960000 ）

The purpose of the present research are;
1) to find an extension of the Atiyah-Singerindex theorem in the framework of Noncommutative Geometry;
2) to apply such anoncommutative index theorem to Geometry and String theory. Achieving the presentresearch we obtained finally the following results.
First we extended the classicalAtiyah-Patodi-Singer index theorem to foliated manifolds with boundary of higherdimensional leaves and obtained an index theorem involved with the Godbillon-Veyclass (a joint work with P. Piazza).
Second, by exploiting the framework of Noncommutative Geometry, we extended the domain of the Godbillon-Vey class (differentiability of foliations), and related our cocycle to the area cocycle defined by T.Tsuboi.
Third, we clarified relationship among the family index theorem of odd dimension, the Dixmier-Douady class (a characteristic class of gerbe) and the Godbillon-Vey class.

Authorship：Collaborating Investigator(s) (not designated on Grant-in-Aid) 

We studied the group of diffeomorphisms which is the automorphism group of the manifold structure. The principal investigator obtained results on the perfectness of the identity component of the group of real-analytic diffeomorphisms,the uniform perfectness and the uniform simplicity of the identity component of the group of smooth diffeomorphisms, and published them. The co-investigators obtained results on the mapping class group of surfaces, on the transversely holomorphic foliations, on connecting lemma in the theory of dynamical systems, and published them. Each year we organized meeting on the diffeomorphism groups for the collaboration and exchange the research ideas.

Authorship：Principal investigator 

Grant amount：\4550000 （ Direct Cost: \3500000 、 Indirect Cost：\1050000 ）

The objective in this project of research is a generalization of the Atiyah-Singer Index Theorem from the viewpoint of Noncommutative Geometry, in situations such as foliated manifolds and manifolds with boundary. We focused on tow notions, the twisted K-theory and the group C*-algebras twisted by cocycles, and sought for Atiyah-Singer type index theorems related to them. We first established a Atiyah-Singer type index theorem for a twisted group C*-algebras of the fundamental group on K-aspherical kaehler manifolds, and obtained certain inequalities on the arithmetic genera of complex manifolds, which is related to the vanishing theorem due to Green and Lazarsfeld. Second we formulated the Atiyha-Patodi-Singer index theorem in the framework of Noncommutative Geometry and extended the theorem in the cases of covering space with infinite degree and foliated manifold with boundary. This formulation provide us with an interpretation of the eta invariant including the higher one, as a pairing of relative cyclic cohomology and relative K-theory, which makes even clearer the geometric significance of eta invariants.

Authorship：Coinvestigator(s) 

Authorship：Coinvestigator(s) 

We obtain many results through the cooperation researches between noncommutative geometry and theoretical physics, namely for proposals on noncommutative gauge theory, p-adic Iwasawa theory, algebraic integrability for quantum Toda lattice, Bernstein measure, string theory and generalized complex structure, quantum cohomology and Frobenius manifolds, symplectic topology and contact topology. The results has been presented by the international conferences and published as monographs in 「Noncommutative Geometry and Physics」, 「Advaned Studies in Pure Mathematics 55」, 「Translations of Mathematical Monographs, 237」 and also published by individual researchers.

Authorship：Coinvestigator(s) 

本研究課題では、変形量子化の収束問題について考察することから幾何学として新しい空間概念であるジャーブが自然に現れることを示し、その性質について研究を行うことを目的としてきた。変形量子化の収束問題を取り扱った研究のなかで、今までにはない新しい解析学の問題や幾何学的として扱う新しい概念が生まれる期待ができてきた。量子的な現象を説明する際に、多様体論の中では説明できない現象が現れてくるように思えることが本研究を行なうための出発点である。実際、変形量子化の収束問題を取り扱うなかで、多様体としては捉えられない空間概念が必要となってくることを示すことができた。このような対象は、積分可能系(特に量子積分可能系)の問題や複素領域での常微分方程式の「動く分岐点」をもつ解空間の問題とも関連することを示すことができた。このなかで、ジャーブ理論を土台に独自の展開を行なうことおこなった。具体的には、1)量子化と非可換解析の整備、2)複素ワイル代数の表現とインタートワイナーの性質、3)非可換指数関数とメタプレクティック群の複素化に対応する幾何学的対象の構成、4)ジャーブカテゴリーでの非線形接続理論と無限小幾何学の設定、5)複素ワイル代数の表現とインターとワイナーの性質、6)非可換指数関数とメタプレクティック群の複素化に対応する幾何学的対象の構成を行ったことである。今までの研究成果をもとにして、幾何学的空間概念の一般化が期待できる。

Authorship：Collaborating Investigator(s) (not designated on Grant-in-Aid) 

Authorship：Coinvestigator(s) 

Gauge theory has still played a central role in low dimensional topology. In particular Seiberg-Witten theory has been studied from the view point of its applications to topology. In this area M. Furuta has introduced a finite dimensional approximation to capture the equation in equivariant homotopy theory, by which he has obtained a refinement of the invariant and the 10/8-inequality for closed spin 4-manifolds.
This research consists of three parts. In the first part we studied this invariant by using homotopy theory. In the second part we gave a geometric interpretation for this invariant. In the third part we studied the index theorem as an application to the geometric quantization conjecture of Guillmen and Sternberg. The detail is given below.
The first one is a joint work with Norihiko Minami. We used a result by Nishida to get a vanishing theorem of the refined Seiberg-Witten invariants for the connected sum of 4-manifolds. In contrasts with this result, we have got a nonvanishing theorem. Then this result tells us that the Seiberg-Witten invariants cannot distinguish differential structure for the connected sum of too many 4-manifolds. In the second one we studied equivariant spin structures on the moduli spaces to Seiberg-Witten equations for a spin 4-manifold. Moreover we extended this result to a spin 4-manifold with positive first Betti number. These imply that we can reprove the 10/8-inequlatiy and its extension from the moduli space with the spin structure. As to the third one, the head investigator got a report that the Guilleman-Sternberg conjecture for Hamitonian torus actions can be reproved from the localization of the Riemann-Roch number.

Authorship：Coinvestigator(s) 

The purpose of the project is to generalize the main result of the joint paper "The Atiyah-Singer index theorem as a passage to classical limit in quantum mechanics" (Com-munications in Mathematical Physics, 182 (1996), 505-533) with G.A. Elliott of the University of Toronto and R. Nest of the University of Copenhagen. In this paper, we studied a certain class of pseudo-differential operators on flat spaces. Employing noncommutative geometric methods we proved an Atiyah-Singer-type index theorem. A crucial property of flat spaces behind the proof is that for given arbitrary two points there exists a unique line segment (geodesic) joining those two points. This property is enjoyed by not only flat spaces but also simply connected hyperbolic spaces, for instance the Poincare disk. Noncommutative geometric methods can be applied to hyperbolic spaces.
The first crucial step to achieve the purpose is to isolate a class of pseudo-differential operators, on hyperbolic spaces, that have Fredholm indices.
In the first year we studied the most important case. That is the Poincare disk. On the flat spaces, the pseudo-differential operators studied are "modelled" on the harmonic oscilators. We constructed the harmonic oscilator on the Poincare disk, what is the Laplacian perturbed by a lower degree term. We showed that the harmonic oscilator, described just above, has compacts resolvents, in particular has a Fredholm index. In the second year, while preparing the paper, a gap in the proof was found, and the most of time was spent on fixing the proof. As a result, unfortunately the goal of the project was not reached in time. The paper on the spectrum of the harmonic oscilator on the Poincare disk will be available some time soon.

Authorship：Principal investigator 

Grant amount：\3400000 （ Direct Cost: \3400000 ）

In the present research we study "Twisted K-theory" and "Twisted Group C*-algebra" and derived the relevant Index Theorem. Twisted K-theory and Twisted Group C^*-algebra have interesting behanior for manifolds with large funcamental groups. Thus it is also interesting to investigate Index theorem on hyperbolic manifolds. Explicitly our objective in this research is stated as follows :
1)We develop the Marcolli-Mathai Index theorem and derive the Index theorem related to Twisted K-theory and Twisted Group C^*-algebras. Also we derive the topological formula for it.
2)We investigate the Index theorem above on hyperbolic manifolds and study the relation to "Geometric secondary invariants such as the Chern-Simons class and R-torsions.
With respect to 1) we clarified the relation among twisted k-theory, Gerbes and the K-group of the twisted groupoid C*-algebras by Cech 2-cocycles with values in U(1). We also developed the twisted Index theorem due to Marcollli-mathai on foliated manifolds and the relevant topological formula. Due to this formula we obtained various interesting results for foliated bundles with large holonomy groups. For instance, when a foliated manifold admits a leafwise symplectic structure and each leaf is K(1)-manifold, then it deoe not admit a longitudinal Riemannian metric with positive scalar curvature. This implies that a generalization of the Gromov-Lawson conjecture still holds for foliated manifolds. Also we proved that Kaehler submanifolds in K-aspherical complex manifolds have non-negative Todd genus up to multiplication of the parity of dimensions.
With respect to 2) we defied the Morita-Hirzebruch invariant on almost contact manifolds and obtained a geometric formula on the eta invariant for 3-dimensional manifolds. Also we clarified the relation among the index theorem for the Reeb vector fields, the Boot localization formula, secondary classes on foliated manifolds and the rotation number of the vector fields due to Ruelle.

Authorship：Coinvestigator(s) 

2006年6月に東京にてPoisson幾何学国際会議が開催されるための準備と調査企画を行った。この国際会議は、1996年ポーランドにて第一回の会議を始めて以来、隔年主に欧米にて開催されてきた。今回、Poisson幾何学の国際的な進展にあわせ、欧米だけではなく日本を中心としたアジア諸国の研究者との融合および討議を目的として開催されるものである。今までの討議の中心が、シンプレクティック幾何学が中心テーマであったものを、その強化とともに、積分可能系、量子化問題、数理物理、特に超弦理論、場の量子論等への拡大を狙った企画が検討されている。欧米が格段に進歩しているシンプレクティック幾何学やポアソン幾何学研究分野の日本を含めたアジア各国の学生および若手研究者の育成にある。国際会議を開催するのにあたり、2005年度に日本およびアジア地域の学生や若手研究者およびその指導者による、研究分野の研究交流を重ねた。このために、欧米への調査派遣、国内において講義、討議を定期的に重ねた。2005年7月にユネスコ、トリエステ(イタリア)で開催されるポアソン幾何学夏の学校は、本国際会議とも連携している。この夏の学校への学生および若手研究者の派遣を行なうことにより、2006年の本国際会議のために若手育成を行なった。このワークショップにおいて、中国、ベトナム、東ヨーロッパ、南米等の若手研究者の教育および来年度に国際会議に招聘する候補者の検討を行った。コロンビアを訪問し、夏の学校での講師、コロンビアにおける若手研究者との交流により、来年度若手研究者を招聘する準備もできた。この国際会議の後援として日本数学会、アメリカ数学会、およびヨーロッパ数学会からスポンサーシップを得ることもできた。ローザンヌ自由工科大学ベルヌーイ研究所等の研究所からの国際連携も行えた。組織委員会、講演者の決定、国際会議のための教育スクールおよび若手研究者の招聘等すべてが整えた。

Authorship：Coinvestigator(s) 

The head Mitsumatsu and an investigator Miyoshi colaborated with others to study the euler class of tangent bundles to foliations and (so called Thurston-Winkelnkemper's) contact structures which are associate with spinnable structures, as a typical class of convergences of contact structures to foliations. Especially they studied the (non-)vanishing of the euler class and the violation of Thurston's inequality and Bennequin's inequlity, from the topological view point of monodromies. As a consequence, a certain class of mapping classes of a surface with boundary can be presented neither as a product of only right-handed Dehn twists nor as that of only right-handed ones. This result was presented in several symposiums including the annual meeting of MSJ in March 2006 as a special invited talk by Miyoshi. The paper is under submission.
The investigator Ono studied the symplectec homology from Floer theory as well as from Seiberg-Witten theory. Including the solution to the Flux conjecture as well as the detemination of the symplectic filling of the link of simple singularities, his contributions to this area are profound.
The investigator Tsuboi studied the relationship between foliation theory and that of contact structures from the view point of the group of contact diffeomorphisms. The investigator Matsumoto stepped further to the foliation theory and studied the ends of Lie foliations.
The head investigator also studied the incompressible fluid dynamics in the framework of the geometry of volume preserving diffeomorphisms and infinite dimensional Hamiltonian systems. He proved that looking from the point of view of such global differential geometry is still valid even to viscous fluides with dissipations. This study is presented in many symposiums, especially as a special project talk in the annual meeting of MSJ in September 2005.

Authorship：Coinvestigator(s) 

We have studied the following :
(1)Study of Yamabe Invariants
We estimated and determined the Yamabe invariant of some positive 3-manifolds, by using the inverse mean curvature flow and families of Green's functions. In especial, we classified completely all 3-manifolds with Yamabe invariant greater than that of RP^3. We also studied the positive Yamabe constants of Riemannian products and the behavior of them under magnifying one factor. We are now studying Aubin's type lemma for the positive Yamabe constants of infinite coverings, with some new results.
(2)Study of Conformal/Affine Geometry and group C^*-algebra
We gave new developments on conformal and projective geometry. In particular, we obtained an interesting varitional characterization of affine connections induced from Einstein metrics. We studied on twisted K-theory and groupoid C^*-algebras, and then proved a generalization of Gromov-Lawson theorem for foliated spaces.
(3)Study on Non-linear Analysis in Geometry
We studied on eigenvalue problem on complete manifolds, discrete groups and valiational problems on mean curvature. We then obtained results on non-existence of eigenvalues on complete manifolds of non-positive curvature, and a fixed-point theorem for discrete-group actions on Hadamard spaces.

Authorship：Coinvestigator(s) 

In Seiberg-Witten theory M.Furuta has introduced a finite dimensional approximation to capture the equation in equivariant homotopy theory, by which he has obtained a refinement of the invariant and the 10/8-inequality for closed spin 4-manifolds.
In this research we improved this inequality by taking into account the quadruple structure on one dimensional cohomology. More precisely we defined a variant of KO-characteristics for closed spin 4-manifoldfs and obtained an additional term determined by this invariant. We also showed that, if the quadruple structure is congruent to the one of 4-dimesional torus or the connected sum of its copies modulo 2, our improvement can be estimated. The researcher was reported by M.Furuta that he is now applying this result to study Seiberg-Witten invariants for symplectic 4-manifolds.
After finishing this work we considered how our result is related to geometry of the moduli space of solutions to the equation. Originally this was studied by P.Kroneheimer, who considered this for low-dimensional moduli spaces. To extend his method to higher dimensional moduli spaces, we introduced a sort of KO-characteristics on the moduli space. Then the 10/8-inequality, as well as the above improvement, can be directly obtained from symmetry of the moduli space with its spin structure. Now we are trying to apply this method in other situations as Yang-Mills theory.

Authorship：Coinvestigator(s) 

This research project aims several problems in Poisson geometry and contact geometry, and quantization problems. Specially, our project focus on the treatments for the quantization problems geometrical point of view. The noncommutative differential geometry is one of our target of our research project. In this project, we had several results on convergent deformation quantization problem. Grebes appears naturally from the construction of the star exponential functions of quadratic forms. Namely, we consider the set of quadratic forms in the complex Weil algebra which forms a Lie algebra isomorphic to sp(n, C). When we consider the esponential functions for these objects, we might expect the complex version of the metaplectic Lie group. Since this is simply connected, we could not handle it. We invested this object by using the explicit computations and have it is in the category of grebes with multiplications. However, it can be described more geometry in terms of the connections. The second problem is to study the invariant deformation quantization problems and construct a convergent star product for ax+b group case. We obtain the universal star product formula. The third result is to study the closed star product. We describe a general settings for obtain how to get the Hochschild cocycle via Stokes formula. This results is still working on and we expect it should be related to the deformation quantization problems for infinite dimensional case. To obtain these results, we have lot of workshops by inviting overseas and domestic researchers together with the research partners. As conclusions, we have fruitful research results which have high evaluation internationally, and also establish the international research network for this area by this grant. This project is still working and will continue for the next project.

Authorship：Coinvestigator(s) 

本研究を主導する研究代表者は、収束する変形量子化の構成から新しい幾何学的概念を展開した。特に、ジャーブ理論との深い関係が解明され、これを詳しく調べた。第二には、Lie環を基本とする1次ポアソン構造、2次ポアソン構造に対する変形量子化を応用できるような具体的体系を整え、その応用へ発展させている。保形形式で注目されているCohen-Rankin積は、不変量子化として考えられているが、この研究を研究分担者である宮崎(琢)の協力を求めて解明をはじめている。森吉は、トポロジーの立場から非可換多様体の不変量の構成、特に指数定理の研究を行なう。特に、非可換トーラス上のDirac作用素による指数定理の構成をめざし、佐々木多様体の指数定理を得ている。亀谷は4次元多様体の不変量として研究が進んでいるザイバーグ・ウィッテン不変量の研究、特に11/8予想についての研究を行って、成果を挙げている。これらの研究の展開のために、国外外研究者との討議等を行ない、海外研究者と研究交流のために、直接海外に赴き、以下の研究者と共同研究や研究討論を行ってきた。
Alan Weinstein : University of California Berkley, Prof., Poisson Geometry
Albert Cattaneo : ETH, Zurich, Prof., Theoretical physics
Alain Connes : IHES, Paris, Prof., Noncommutative geometry
その成果をまとめるために、A.WeinsteinとA.Cattaneoが来年来日する。

Authorship：Coinvestigator(s) 

In this research project, we have introduced a new topological invariant, called the "block number", for 3-manifolds, which estimates some kind of complexity of a 3-manifold just like as the Heegaard genus. The block number is defined by means of a flow-spine, and is enable us to classify 3-manifolds, and to parameterize 3-manifolds in each class by finitely many integers. Moreover the block number can be defined not only for a 3-manifold but also for a combed 3-manifold, a pair of a 3-manifold and a non-singular vector field on it. Using this invariant, we have gotten the following results :
1.The only combed 3-manifold having 0 as its block number is the canonical one on the product of the 2-sphere and the circle, and combed 3-manifolds with the block number 1 are canonical ones on lens spaces (including the 3-sphere).
2.The parameterization for 3-manifolds with the block number 2 was given. And, using the Reidemeister torsion, we have shown some results which imply that our parameterization uniformize the presentation of a combed 3-manifold.00
3.We have given a formula for calculating the value of the Thraev-Viro invariant for all Seifert fibered 3-manifolds.
On symplectic manifolds, we have gotten the following result :
4.If the universal covering space of a clsed symplectic manifold is contractible, the manifold does not admit any Riemannian metric with positive curvature.

Authorship：Coinvestigator(s) 

The purpose of this project is to obtain a quantum version of the results in "The Godbillon-Vey cyclic cocycle and longitudinal Dirac operators (with the investigator Hitoshi Moriyoshi)" and "Topological approach to quantum surfaces( with Ryszard Nest of the University of Copenhagen)", more precisely to construct noncommutative Anosov foliations on "the unit tangent bundles" over noncommutative Riemann surfaces. This noncommutative Anosov foliations are regarded as quantizations of the (commutative) Anosov foliations associated with geodesic flows on the unit tangent bundles. The ultimate goal of the project is to prove the foliation index theorem of A. Connes, for the noncommutative Anosov foliations.
In a joint project with Nest (unpublished) we constructed noncommutative 3-manifolds as strict quantizations of unit circle bundles of closed Riemann surfaces of genus greater than 1.These noncommutaive 3-manifolds were constructed in such a way that the relationship between the Riemann surface and its unit tangent bundle is kept intact through a suitable group action. Moreover we constructed a foliation on the noncommutaive 3-manifold as a certain C^*-algebra in the spirit of A. Connes's noncommutative geometry. We are preparing a paper "Noncommutaive Anosov foliations (tentative title)". We are currently working on detail. As one expects, on view of commutative case, the C^*-algebra representing a "leaf of the noncommutaive foliation is a covering space. We developed some idea how to lift the Dirac operator on the quantized Riemann surface to a longitudinal elliptic operator for the noncommutative Anosov foliation.
Unfortunately we were unable to complete the project. However, we certainly continue to work on the project, as we now have a clear idea how to achieve the goal.

Authorship：Principal investigator 

Grant amount：\3300000 （ Direct Cost: \3300000 ）

In this research we proceeded to a generalization of the Atiyah-Singer index theorem on the basis of Low-dimensional Topology. Here we state one of the results of the project, which is related to the Atiyah-Patodi-Singe Index Theorem in Non-commutative Geometry.
Let Γ be a discrete group and σ a 2-cocycle of Γ with values in U(1). We then twist the product on the group algebra C(Γ) in the following way : U_gU_h = σ(g,h)U_<gh> where U_g, U_h are the formal unitary elements corresponding to g, h ∈ Γ. Due to the cocycle condition we obtain an associative product on C(Γ). With respect to the operator norm on L^2(Γ) we take the C^*-closure of C(Γ) with product above. It is called group C^*-algebra twisted by a 2-cocycle σ and denoted by C^* (Γ,σ).
There exists a Dirac operator whose index belongs to the K-group of C^* (Γ,σ). Let us denote the Dirac operator by D^▽. We then obtain the Index formula which express the trace τ(IndD^▽) of the index IndD^▽ in trems of characteristic classes A^^^(M/Γ) and a curvature R of the associated line bundle. As a corollary of the formula. we can prove the following result :
Suppose that a closed symplectic manifold M is aspherical, then M does not admit a Riemanninan metric of positive scalar curvature. This yields a prtial solution to the Gromov-Lawson conjecture.

Authorship：Coinvestigator(s) 

本研究は、非可換幾何学を中心として数理物理学への応用を図るために、特に国外研究者や国外研究機関との連携を強くする目的で行われた。その理由は、本研究およびその関連分野は国内において残念ながらあまり大きな進展が行われていないにもかかわらず、国際的には、数学の重点研究領域になっているからである。一方、数理物理学、素粒子論への深い関わりがあり、それらの関係と大きな接点を求め研究を展開した。第一には、変形量子化問題の収束性に精力を注いだ。形式的な変形量子化問題については、ある程度の結果が得られているが、収束性については、さまざまな現象が起こりその解明を行うことが優先であった。本研究の討論を通して、新しい幾何学的概念として、Blured manifoldsを提唱することが今後の課題であるという評価を得ることができた。本研究に関連した研究は欧州が主力となっており、英国を中心とした研究交流を盛んに行った。特に、日英ウインタースクールを開催し、日本から若手研究者を中心とした参加者を募り、英国、フランスおよび周辺各国からの関連研究者を集めたシリーズレクチャーと一般講演を行った。これには、総勢60名以上の参加を得た。そのほか、2002年11月には東北大学にて、超弦理論と非可換幾何学に関する国際ワークショップを開催し、これも国内外総勢60名以上の参加者を集め、研究討論が行われた。以上、本研究は、外国人招聘、国外研究機関での討論および国際ワークショップを中心とした、国内外の研究者交流を行い、将来的に2004年に国際会議の開催を目指すための準備が整えた。

Authorship：Coinvestigator(s) 

Based on the notion of asymptotic linking, the head investigator proposed the framework where the study on contact structures and that on foliations would be unified, and began the research. Op the side of foliations, it turned out that exotic classes and the 1st foliated cohomology are strongly related to this framework. On the other side, the torsion invariant of contact structures has a deep relation with it. For algebraic Anosov foliations, we also established the computation of its 1st foliated cohomology and found its relation to local orbit rigidity.
A research group including Miyoshi and Mitsumatsu investigated Thurston's inequality for foliations on the boundary of compact Stein surfaces and established the absolute version of the inequality for certain cases. The relative ersion and more general case are left for the future as important subject. Tsuboi and Mitsumatsu worked on the perfectness of groups of diffeomorphisms preservein certain geometric structures. Especially Tsuboi provednthe perfectness for contact diffeomorphisms and analytic diffeomorphisms of certain manifolds. Tsuboi also classified regular bi-contact structures on Seifert fibered spaces.
Another group including Ono and Ohta, mainly working on contact/symlectic topology, characterized the symplectic diffeo-types of the filling of the link of simple and hyper-elliptic singularities. Also they got started the construction of obstruction theory for Lagrangian Floer homology theory.

Authorship：Coinvestigator(s) 

In this project, we focussed on the study of the structure of the mapping class group of surfaces (m.c.g. for short) as well as the moduli space of compact Riemann surfaces, together with various problems closely related with this. They include the following thema : cohomology group of m.c.g., the theory of the Floer homotopy types, topological invariants based on gauge theory, construction of the harmonic Magnus expansion of m.c.g., structure of the Grothendieck-Teichm＼"uller group, the volume conjecture, non-commutative geometry in dimensions 3,4, finite subgroups of m.c.g., the Jones representation of m.c.g., relation between m.c.g. with 4-dimensional topology. From the interactions of these thema, we found new directions of research such as the relation between the geometry of m.c.g. and the symplectic topology as well as the comparaison between m.c.g. and the outer automorphism group of free groups.

Authorship：Coinvestigator(s) 

The aim of the project is to give a constructive proof of existence of analytic deformation of Poisson manifolds, that generalize symplectic manifolds. The existence of deformation quantization(algebraic deformation) for Poisson manifolds, which has long been an important problem, was finally shown by M.Kontsevich in 1997. The relationship between algebraic deformation and analytic deformation is similar to the relationship between a formal power series and a smooth function that realizes the given formal power series.
Symplectic manifolds are special examples of Poisson manifolds, and its structures are well known. In a joint project with R. Nest of the University of Copenhagen and I.Peter of Munster University, we investigated symplectic manifolds and showed that any closed symplectic manifold has an analytic deformation provided that the second homotopy group is trivial. This result is published as "Strict quantization of symplectic manifolds (to appear in Letters hi Mathematical Physics)".
The second homotopy group of the 2-sphere is nontrivial. Thus the result above cannot be applied to the 2-sphere. In a joint project with C.L.Olsen of the State University of New York at Buffalo, we studied the 2-sphere. The 2-sphere possesses interesting Poisson structures besides the standard rotation invariant symplectic structure. We constructed an analytic deformation for a Poisson structure degenerate at the North and South poles. This result is published as "A new family of noncommutative 2-sphers".

Authorship：Coinvestigator(s) 

Poisson Geometry, which is the extended notion of symplectic geometry, is now developing quite recently. However, this research fields is not common in Japan, while this research field is much developing in Europe and the United State. In our Study, we have a mission to develop this research fields in Japan, and to coorporate with the researchers who are working on this fields in Europe and the Unites States. The first our development is to study the convergence problems on deformation quantizations. For the case of formal deformations, M. Kontsevich has given a pretty result. It is important problem to study the convergence for the deformation quantizations. Through our research, we found the totally different feature for the convergence of the deformation quantizations, and propose a new geometric objects on gerbes. We will extend our research on the convergence of the deformation quantization as a future task. By this grant, we have two major international symposia in 2001 and 2002, which has noncommutative geometry and D-brane as main Topics. We could have many visitors from abroad and also from domestic research institutes. We have published a proceedings for our research developments. Maeda has been invited to the Poisson 2002 international conference at Lisbon and gave a talk on this problem, which was very interesting for the participants. We have also visited various meetings in Japan and outside of Japan to make strong activities. As a results, we could have a international research groups on noncommutative geometry, which is able to develop the research.

Authorship：Principal investigator 

Grant amount：\3400000 （ Direct Cost: \3400000 ）

In this research we proceeded to a generalization of the Atiyah-Singer index theorem on the bas1S of Low dimensional Topology. The objective of the project are the followings:
1. Establish the elaborated Index Theorem in the framework of Non-commutative Geometry. Also study the relationship between the Index Theorem and the analytic secondary invariants like the eta invariants and the spectral flow;
2. Study the relationship between the elaborated Index Theorem and secondary characteristic classes including the Maslov class.
Here we state one of the results of the project, which is related to the Atiyah-Patodi-Singe Index Theorem in Non-commutative Geometry. In July 2001, we are invited to give a talk with the title 'Analytic K-theory and the index theorem' at Topology Symposium of Japan, Akita. We are also working on the project are presented with the title 'Eta invariants, the Godbillon-Vey classes and the index theorem' in March 2001 at Workshop on Non-commutative Geometry and String Theory, Keio University. We also developed the research on Non-commutative Hopf invariants, and presented some results at Nagoya Technology Institute with the title 'Spectral flow, the Teoplitz index and the Hopf invariant' in March 2001, and with the title 'A index theorem of vector field and the Hopf invariant' at Kagoshima University in February 2002.

Authorship：Coinvestigator(s) 

In a joint project with R.Nest of the University of Copenhagen and I.Peter of the University of Munster the pricipal investigator showed that under a topological condition every closed symplectic manifold has a strict quantization. Strict quantization is an analytic deformation theory. An algebraic deformation theory (existence of deformation quantization) has been known since 80's.
The aim of the project is to show existence of strict quantizations for Poisson manifolds, that generalize symplectic manifolds. The existence of deformation quantization for Poisson manifolds, which has long been an important problem, was finally shown by M.Kontsevich in 1997. The project is divided into three steps. The first step is to re-examine the existence proof of strict quantization for symplectic manifolds, in order to have a deep understanding of mechanism of existence. In particular, re-examination of the proof by B.Fedosov, which played a crucial role in our proof, of existence of deformation quantization is an important step. The second step is to understand the existence proof of deformation quantization for Poisson manifolds and to rewrite it from the viewpoint of Fedosov. The last step involves actual construction of strict quantization.
Through quite a few discussions with Nest, the mechanism of existence became fairly clear, and we obtained a refined version of our result. Thanks to a recent appearance of a simpler proof of existence of deformation quantization for Poisson manifolds than Kontsevich's, we have a prospect to achieve the second step.
While working on the project discussed above, in a joint project with C.L.Olsen of the State University of New York at Buffalo, the principal investigator worked on the cases that are not covered by the results with Nest and Peter. In particular, we showed that the 2-sphere with a specific Poisson structure has a strict quantization. In the process to construct strict quantization we obtained new "noncommutative 2-spheres". These C^*-algebras are new examples of noncommutative Poisson manifolds.
As explained above, unfortunately we could not achieve the goal of the project, i.e. the existence of strict quantizations for poisson manifolds. We certainly intend to continue working on the project. We will hopefully complete the project within a year or so.

Authorship：Coinvestigator(s) 

1. Multiplicity of recurrence of type II ∞ ergodic transformations : We classified the set of Markov shifts by the return sequence and also by Kakutani-Parry index. This result was appeared in a paper by J.Aaronson and H.Nakada, Israel Journal of Math. 2000. Moreover, Hamachi's research group has shown that the multiplicity of recurrence is preserved by the compact group extensions.
2. It has been known that the cardinality of the set of locally finite ergodic nvariant measures for a cylinder flow is continuous if the rotaion number of the base transformation is of bounded type. These measure are induced from the PL homeomorphisms of the circle, those were considered by Herman. In this project, we proved that there is no other locally finite invariant measure for such cylinder flows. On the other hand, we considered Maharam extensions of adding machines of Markovian type. We also determined the set of locally finite ergodic invariant measures for such Maharam extensions associated to Hoelder continuous potentials.
3. We studied continued fraction expansions of formal power series with a finite field coefficients. Moreover we considered the metrical theory of diopantine approximation in positive characteristic. We showed that analogue of some classical metric theorems hold. In particular, we proved the formal power series version of Dufine-Schaeffer thoerem.

Authorship：Coinvestigator(s) 

We have investigated the structure of the mapping class group of surfaces as well as the geometry of moduli space of Riemann surfaces mainly from the viewpoints of topology. The main results we obtained are as follows.
(i) The subalgebra of the rational cohomology algebra of the mapping class group of surfaces generated by the Mumford-Morita classes is called the tautological algebra. There have been three approaches to the study of this tautological algebra. The first is based on the twisted Mumford-Morita classes introduced by Kawazumi, the second is in terms of invariants for trivalent graphs and the third is through symplectic representation theory. Summarizing previous results, we found that the above three approaches correspond exactly to each others.
(ii) The theory of secondary characteristic classes of the mapping class group is still a largely unknown area. However, in this research, we proved that these secondary characteristic classes have deep structures that cannot be detected by the nilpotent completion of the Torelli group. It seems highly likely that the solvable or semi-simple structure of the Torelli group will become more and more important in the future.
(iii) We made significant progress in understanding the algebro-geometrical as well as the topological structure of families of Riemann surfaces. In particular, we obtained many results concerning the monodromies of symplectic fibrations.

Authorship：Coinvestigator(s) 

(1) On Ds-diagrams
It was shown that two Ds-diagrams representing the same closed 3-manifold can be transformed to each other by a sequence of "elementary deformations".
(2) On Heegaard splittings
A new condition for a Heegaard splitting to be reducible has been found. This condition is described by the notion of a "d-pseudo core".
(3) On framed links in a homotopy 3-sphere
It was shown that any homotopy 3-sphere admits a "very special framed link" , which enjoys a good property and closely related to a Heegaard splitting.
(4) On the Poincare conjecture
Using the above two results (2) and (3), we have proposed a new method for attacking the famous "Poincare conjecture".

Authorship：Coinvestigator(s) 

シンプレクティック幾何学の今後の方向性・展望を、他の分野との関連を含めて明らかにすることを意図し、シンプレクティック幾何学全般、接触幾何学、代数幾何学、ケーラー幾何学、大域解析学、非可換微分幾何学、ゲージ理論、場の量子論の諸立場から、企画・調査・検討を行なった。「企画調査」ということなので、オリジナルな結果を探るという形ではなく、この分野に対する新たな基礎づけや、位置づけ・動機づけを与えることを考えた。これに関して、分担者のうちの数人がグループを組んで、あるいは個々別々に、とさまざまな分野の専門家たちと各地で交わる機会を持ったことは有意義だったと考える。シンプレクティック幾何学およびシンプレクティック・トポロジーをテーマとする研究集会・セミナー等が来年度以降多く催されるが、その後を引き継ぐテーマをいくつか得ることができた。具体的には、小野氏・太田氏からは特異点理論との接点を、三松氏・森吉氏からは変形量子化と低次元多様体の不変量の関連が、それぞれ提案され、詳細な報告と吟味がなされた。これからのシンポジウム等の企画に直結させることができると信じる。同時に、森吉氏のSurveys in Geometryにおける講演等、本研究組織のメンバーが各地の集会において総合報告をする機会を多く持つことができたことも実績の一つとして挙げたい。
一方、海外の研究者との研究連絡・打ち合わせを小野・太田両氏が行ない、今後の国際交流の推進を計るとともに、国内では、三松佳彦氏が研究代表者を務める“Encounter with Mathematics(数学との遭遇)"と連携することを通して、学部生・大学院生等に接する機会を多く持ったことも、本研究組織の活動として評価したい。

Authorship：Principal investigator 

Grant amount：\3100000 （ Direct Cost: \3100000 ）

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.

Authorship：Coinvestigator(s) 

For "classical theory" we have obtained two major results. One of them may be stated as follows : Let M be a riemannian manifold diffeomorphic to 2-sphere, and let F be a first integral of its geodesic flow that is a polynomial of degree kappa on each fiber. Such a pair (M, F) is well-understood if kappa = 1,2. When kappa <greater than or equal> 3, however, no nontrivial example has been known except one-parameter families for kappa 3 and 4. In this research we constructed families of such (M, F) (parametrized by functions in one variable) for every kappa <greater than or equal> 3. Moreover, we proved that they are C_<2pi>-manifolds, i.e., every geodesic is closed and has the common length 2pi. The other result is a construction of "Hermite-Liouville structure" on Hopf surfaces. The idea is analogous to Kahler-Liouville manifold, which is a complexifled version of Liouville manifold established by the head investigator. Despite the significance of the Kahler condition in the whole theory of Kahler-Liouville manifolds, this result seems to suggest the existence of another complexification scheme for Liouvil
For "quantization" we studied spectra of the laplacian on Liouville surfaces diffeomorphic to 2-sphere. We decomposed the defining equation of the eigenfunctions into a pair of ordinary differential equations on circles, and applied semiclassical approximation to each of them. As a result, we found that this method gives a nice approximation when the corresponding invariant tori are sufficiently close to a critical one, as well as the case where the tori are located far from the critical ones. We think this result will be more refined.

Authorship：Coinvestigator(s) 

1997年から1998年にかけては,次の研究テーマを主題に行なった.
(1) 4次元空間のYang-Mills gradient flowと3次元空間のYang-Mills-Higgs gradient flowの研究.
(2) 変形量子化問題と非可環幾何学
(3) 無限次元空間における無限次元リー群の作用によるオービットの幾何学的性質
(1)については,4次元コンパクト多様体の
Yang-Mills qradient flowについて,チャーン数に近い初期データを与えることでその滑らかな解が大域的に存在することを示した.また,3次元空間のYang-Mills-Higgs qradient flowについては,その弱大域解の存在を与えられることがわかった.
(2)については,特に接触多様体の変形量子化問題とそのレダクションの方法について研究を行なった.
(3)についてはコンパクト多様体のリーマン計量の空間に作用する微分同型群(無限次元リー群)のオービットに対する平均曲率の定式化とその応用について研究を行なった.
これらの研究は萌芽研究として申請した,非可換微分幾何学の構築において基礎となる成果をあげることができた.そして,その成果はこの2年間の間に行なわれた,国内の研究集会,国際研究集会で発表や討議を行ない,これから先に非可換微分幾何学の展開に向けて,大きな成果をあげた.さらなる成果は,近い将来に出版される予定である.

Authorship：Coinvestigator(s) 

The summary of Research Results is as follows. The head investigator studied the correspondence between the equivalence problem of the systems of linear differential equations of finite type (Holonomic system) and that of the projective embedding images of their fundamental solutions (projective solution). As the application of Seashi's theory for the former problem, we showed that the image of the projective solution of the Hyper geometric system E(n, k) does not lie in the Grassmannian manifold Gr(k-1 , n-1) (the image of Plucker embedding) except for E(3,6). Furthermore we discussed the generalization of E.Cartan's paper on 5 variables.
Izumiya classified the generic singularities of solution surfaces forquasi-linear first order partial differential equations and also classified the generic bifurcation of viscosity solutions for Hamilton-Jacobi equations of space dimension 1.
Kiyohara defined the notion of Liouville manifolds and Kahler-Liouville manifolds, which are two classes of riemannian manifolds whose geodesic flows are integrable and studied their structures in detail. He carried out a part of classification and found out a new family of so-called "Cl-metrics" on the Sphere.
Ishikawa showed the transversality theorem of Thom-Mather type for the space of isotropic mappings of corank 1 into symplectic manifolds and gave the characterization of Mather type or Arnold type for the Symplectic stability and Lagrange stability of the isotropic mappings.
Kawazumi developed new tools to calculate the cohomology groups over finite fields of the hyperelliptic mapping class groups and gave an overview on the study of the cohomology groups of the moduli spaces of Riemann surfaces by the complex analytic Gelfond-Fuchs cohomology.

Authorship：Principal investigator 

Grant amount：\1200000 （ Direct Cost: \1200000 ）

本研究では、
1.“Spectral flow"あるいは“Eta-invariant"といった解析的二次不変量を捉えるために、K-理論や巡回コホモロジー理論を表現のための主要な手段としながら,二次特性類が関与する精密化された指数定理を構築すること;
2.この精密化された指数定理をシンプレクティック幾何学の範疇で考察し、そこで二次特性類として現われてくるMaslov類との関連を明らかにすること;
を主要な研究目標とした。また主要な実例がS^1の微分同相群の等質空間や無限次元グラスマン多様体などの不変微分型式と密接に関連していることに着目して、これらの空間のシンプレクティック型式や微分同相群から派生する二次特性類とMaslov類との関連性について研究をおこなった。
本年度における具体的な結果としては、Maslov類およびSpectral flowとユニタリー群の中心Z拡大との関連性を明確にしたことが挙げられる。この結果については1997年4月の日本数学会特別講演において発表予定である。またPacific Journal誌に掲載された論文:S^1の場合についての主要な二次特性類であるGodbillon-Vey類に対するAtiyah-Singerの指数定理の一般化(これは「巡回コホモロジー群による指数定理の一般化」という昨年度の奨励研究Aの課題と密接に関連する)も、二次特性類が関与する精密化された指数定理の構築に向けて得られた一つの結果である。

Authorship：Principal investigator 

Grant amount：\800000 （ Direct Cost: \800000 ）

本研究では、Connesによる非可換微分幾何の枠組に基いてAtiyah-Singerの指数定理を捉え直すことを目的とした。特に、葉層束に関して二次特性類が関与する巡回コサイクルを構成すること、およびこのような巡回コサイクルとK-群の元との対合を考察しAtiyah-Singerの指数定理の一般化を導くことに研究の重点をおき、さらに主要な実例に現れるこのような巡回コサイクルに関して、無限次元の等質空間の不変微分型式(例えば1次元球面の微分同相群の等質空間あるいは無限次元グラスマン多様体上のシンプレクティック型式)との関連性を考察した。またK-群や巡回コホモロジー理論とSpectral flowやEta-invariantといった解析的二次不変量と二次特性類が関与する精密化された指数定理に関する考察もおこなった。
本年度における具体的な結果としては、S^1の場合についての主要な二次特性類であるGodbillon-Vey類に対するAtiyah-Singerの指数定理の一般化(これはProceedings of“Geometric Study of Foliation"(1994)に掲載された昨年度の研究結果と前後して密接に関連する)が得られた。これは閉曲面上の葉層S^1束に対するGodbillon-Vey数が、非可換微分幾何における「曲率」と考えられることを示しており、本研究の目的に照らして満足すべきものと思われる。この結果はPacific Jurnal誌に掲載予定である。

Authorship：Principal investigator 

Grant amount：\1200000 （ Direct Cost: \1200000 ）

本研究では、多様体上に定義される上記のような幾何的な作用素全体の成す空間が自然に無限次元グラスマン多様体に埋め込まれていることに着目し、無限次元グラスマン多様体と作用素全体の成す空間の位相的な関連、そして作用素全体の空間の構造が初めに与えられた多様体の位相をどのように反映しているかについて調べることを目的とした。とくに作用素全体の成す空間には多様体の微分同相群が作用していることから、微分同相群のコホモロジー類あるいはその離散部分群から構成される葉層束の二次特性類との密接な関連が予期されるので、その関連を明確にすることに研究の重点をおいた。
本年度の研究においてはS^1の場合について主要な二次特性類であるGodbillon-Vey類およびEuler類と、S^1の微分同相群全体のなす空間から無限次元グラスマン多様体への埋め込みから引き戻しとして得られる微分同相群の等質空間上のシンプルレクティック構造との関連が明確にされた(この結果はComtemporary Math.誌に掲載予定である)。さらにこのようなS^1の場合の具体例をふまえて、高次元の葉層束のGodbillon-Vey類に対するConnesにより提唱された非可換微分幾何学の枠組に沿う結果をも得た(これはProceedings of "Genometric Study of Foliation"に掲載された)。