Applied singularity theory to exterior differential systems

将奇点理论应用于外微分系统

• 批准号: 14340020
• 负责人: ISHIKAWA Goo
• 金额: $ 7.62万
• 依托单位: HOKKAIDO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2002
• 资助国家: 日本
• 起止时间: 2002 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14340020/
• 关键词: Monge-Ampere equation; Goursat distribution; singular Legendre submanifolds; coisotropic mapping; symplectic moduli space; singular Legendre knot; mapping space quotient; uni-modal plane curve; ガウス・クロネッカー曲率; 接触モデュライ空間; ユニモダル特異点; モンジュ・アンペール系; ア

The following results are obtained : Classification of generic singularities in geometric solutions to Monge-Ampere equations. Clarification of Goursat-Legendre correspondence. Basic investigations on singular Legendre submanifolds, in particular singular Legendre curves. Bifurcation of singularities of developables. Estimate on doubly degenerate submanifolds under projective duality. Introduction of the notion of singular coisotropic mappings. Proof of the localization theorem on the symplectic moduli spaces. Basic study on mapping space quotients. Classification theory of singular Legendre knots. Discovery of new singularities of solutions to Monge-Ampere equations in dimension three. Classification of uni-modal curve singularities and determination of their symplectic moduli spaces. Discovery of remarkable similarity between diffeomorphism classification of plane curve singularities and contactomorphism classification of associated Legendre curve singularities.

得到了以下结果：Monge-Ampere方程几何解中类属奇点的分类。Goursat-Legendre对应关系的澄清。奇异勒让德子流形的基本研究，特别是奇异勒让德曲线。可展项的奇点分支。投射对偶下双退化子流形的估计。引进奇异余迷向映射的概念。辛模空间上局部化定理的证明。空间映射的基础研究。奇异勒让德纽结的分类理论。发现三维Monge-Ampere方程解的新奇点。单峰曲线奇点的分类及其辛模空间的确定。发现了平面曲线奇点的自同构分类与伴随Legendre曲线奇点的接触自同构分类之间的显著相似性。

项目成果

Symplectic Bifurcations of Plane Curves and Isotropic Liftings

• DOI: 10.1093/qjmath/54.1.73
• 发表时间: 2003-03
• 期刊: Quarterly Journal of Mathematics
• 影响因子: 0.7
• 作者: G. Ishikawa;S. Janeczko

Perturbations of Caustics and Fronts, Vol.62

焦散和前沿的扰动，第 62 卷

• 发表时间: 2004
• 作者: Masatoshi KOKUBU;Masaaki UMEHARA;Kotaro YAMADA;G.Ishikawa
• 通讯作者: G.Ishikawa

G.Ishikawa, M.Kimura, R.Miyaoka: "Submanifolds with degenerate Gauss mappings in spheres"Advanced Study in Pure Mathematics. 37. 115-149 (2002)

G.Ishikawa、M.Kimura、R.Miyaoka：“球体中具有简并高斯映射的子流形”纯数学高级研究。

Equivariant Chern classes for singular algebraic varieties with group actions

具有群作用的奇异代数簇的等变陈氏类

• 发表时间: 2005
• 期刊: Math.Proc.Cambridge Phil.Soc. 139(印刷中)
• 作者: F.Aicardi;T.Ohmoto;Toru Ohmoto
• 通讯作者: Toru Ohmoto

T.Ohmoto: "Self-intersection class for singularities and its application to fold maps"Transaction of American Mathematical Society. 355-9. 3825-3838 (2003)

T.Ohmoto：“奇点的自交类及其在折叠图中的应用”美国数学会汇刊。