Project/Area Number  06452008 
Research Category 
GrantinAid for Scientific Research (B).

Research Field 
Geometry

Research Institution  Kobe University 
Principal Investigator 
SASAKI T Kobe University, Dept Math, Professor, 理学部, 教授 (00022682)

CoInvestigator(Kenkyūbuntansha) 
浜畑 芳紀 神戸大学, 自然科学研究科, 助手 (90260645)
樋口 保成 神戸大学, 理学部, 教授 (60112075)
TAKANO K Kobe University, Dept Math, Professor, 理学部, 教授 (10011678)
TAKAYAMA N Kobe University, Dept Math, Associate Professor, 理学部, 助教授 (30188099)
KABEYA Y Kobe University, Dept Math, Assistant, 理学部, 助手 (70252757)
IKEDA H Kobe University, Dept Math, Professor, 理学部, 教授 (10031353)
NAKANISHI Y Kobe University, Dept Math, Associate Professor, 理学部, 助教授 (70183514)

Project Fiscal Year 
1994 – 1995

Project Status 
Completed(Fiscal Year 1995)

Budget Amount *help 
¥4,600,000 (Direct Cost : ¥4,600,000)
Fiscal Year 1995 : ¥1,700,000 (Direct Cost : ¥1,700,000)
Fiscal Year 1994 : ¥2,900,000 (Direct Cost : ¥2,900,000)

Keywords  projective invariant / hypergeometric system / regular graph / Painleve system / radial solution / unknotting number / Groebner basis / 射影不変量 / 超幾何微分方程式 / 正則グラフ / パンルベ系 / 球対称性 / 結び目解消数 / グレプナ基底 / 変曲点 / アファイン極小性 / パンルベ方程式 / モジュラー曲面 / ロ加群 / 球対称解 
Research Abstract 
We give an outline of the results on the project. (1) On the geometric structures in affine and projective differential geometry and its equivalence problem. 1. We gave an estimate on the number of inflection points from below that are lying on a closed curve on flat affine torus and examples that show this estimate is the best. With this estimate we could characterize the Euclidean torus among flat affine tori. 2. We proved that mdimensional submanifold in the affine space of dimension m+m (m+1) /2 has a canonical connection and characterized the affine minimality of such submanifolds. 3. We published a book on affine differential geometry that gives the foundation of such geometry. 4. We developed the projective theory of submanifolds and classified projectively homogeneous surfaces. (2) On the geometric structures associated with the system of hypergeometric differential equations. 1. We proved that the projective mapping of the hypergeometric differential system denoted by E (k, n)
… More
does not have generally its image on any Grassmannian submanifolds. 2. The configuration space of 5 points on a projective line can be described combinatorially by decomposing the space into 20 simply connected domains. 3. We defined the system associated with the integral on the configuration space of one conic and any number of hyperplanes on the projective space, clarified its symmetry, the contiguity relations, and computed its intersection form. 4. We proposed a new algorithm for integer programming by applying contiguity relations of the system E (k, n) and proved related results. 5. We decided the general formulae of contiguity relations for the confluent hypergeometric sytems. 6. We succeeded in the description of Stokes phenomena by use of Braid group. (3) On the geometric structure of Painleve system. 1. We poved that the space of initial states for Painleve system has a symplectic structure and provided a new method of the characterization of such a system. 2. We solved the reduction problem for the Garnier system. (4) On the deformation equation of conformal structures. 1. We investigated a global structure of the nonlinear differential equation associated with the deformation of conformal structures. 2. We proved the existence and the uniqueness of radial solutions for this equation. (5) Studies on combinatorics and algorithms. 1. We gave a growth estimate of the coefficients of Alexander polynomials of knot and applied it to the linkhomotopy theory. 2. We gave a partial answer to the conjecture related with minimization of the unknotting number. 3. We gave a method of knowing local equivalence in terms of generalized unknotting procedure. 4. We clarified the relation of deltaunkotting number with Conway polynomials. 5. We provided an algorithm that enables the computation of binomial relations. 6. We improved the mathematical processing system KAN and implemented it to Macaulay. 7. We developed an algorithm for Voronoi diagram on the negativelycurved plane and applied it to information geometry. (4)共形構造の変形問題について 1.幾何構造の中で非線形微分方程式と関連した重要な対称に共形構造の変形問題の大域的研究 2.共形変形に関連した非線形楕円型方程式の球対称解の存在、一意性等についての研究 (5)組み合わせおよびアルゴリズムの研究 1.結び目に対するAlexander多項式の係数の増大の仕方についての評価式およびそのlinkhomotopyへの応用 2.結び目解消数の最小化についてのある予想への部分的解答 3.一般化した結び目解消操作による結び目の局所同値による分類 4. Deltaunknotting numberがtorus knots, positive pretzel knots, positive closed S4Sbraidsなどのクラスで、Conway多項式の2次の係数と一致することの証明 5. 2項係数の和のみたす漸化式を計算するアルゴリズムの作成 6.数式処理システムKANの充実、Macaulayなどへの実験的組み込み 7. 2次元の負定曲率空間でVoronoi図を書くためのアルゴリズムの開発とその情報幾何への応用の研究 Less
