Toward construction of the universal complex of the solutions of differential equations

微分方程解的通用复形的构建

• 批准号: 21740056
• 负责人: YAMAMOTO Takahiro
• 金额: $ 2.83万
• 依托单位: Kyushu Sangyo University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Young Scientists (B)
• 财政年份: 2009
• 资助国家: 日本
• 起止时间: 2009 至 2012
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-21740056/
• 关键词: ジェネリックな写像; 特異点論; 微分方程式; 特異点集合; ホモトピー類; 発散図式; カスプ特異点; 常微分方程式; 安定写像; 特異値集合

1st order differential equations are considered as hypersurfaces in the 1st jet space and solutions of the 1st order differential equations are curves on the hypersurfaces. Since, the classification of divergence diagrams induces the classifications of a certaintype 1st order differential equations, we study smooth maps between surfaces. It is well known that a smooth map between surface is approximated by a smooth map, which is called a generic map, whose singular values set is a regular curve with some cusps. For a smooth map, generic maps which are homotopic to the smooth map and whose singular values setare the simplest, in some sense, are studied. Furthermore, for a smooth map between surface, the pairs of the numbers of singularities, cups and nodes, on the singular values set of generic maps homotopic to the smooth map are determined.

将一阶微分方程解看作一阶喷流空间中的超曲面，一阶微分方程解为超曲面上的曲线。由于散度图的分类归结为一类一阶微分方程组的分类，因此我们研究曲面之间的光滑映射。众所周知，曲面之间的光滑映射是用光滑映射来逼近的，这种映射称为广义映射，它的奇异值集是一条具有某些尖点的正则曲线。对于光滑映射，研究了与光滑映射同伦且奇异值集在某种意义上是最简单的一般映射。此外，对于曲面之间的光滑映射，确定了与该光滑映射同伦的一般映射的奇异值集上的奇点个数对，即杯数和节点数。

项目成果

Survey of apparent contours of stable maps between surfaces

表面之间稳定地图的表观轮廓的调查

• 期刊: Proceedings of The 6th Franco-Japanese Symposium on Singularitiesに掲載決定済
• 作者: 亀之園淳;山本卓宏;Minoru YAMAMOTO;Yuuki Tadokoro;Akira Sakai;A. Shioura;川上裕・中條大介;T.Yamamoto
• 通讯作者: T.Yamamoto

Topology of the singular value sets of stable maps

稳定映射奇异值集的拓扑

• 发表时间: 2012
• 作者: N. Shakhlevich;A. Shioura;V. Strusevich;山本稔・平澤美可三;田所勇樹;山本卓宏
• 通讯作者: 山本卓宏

Apparent Contours of Stable Maps into the torus

环面稳定贴图的明显轮廓

• 发表时间: 2009
• 作者: Vladimir Kolmogorov;Akiyoshi Shioura;山本稔;山 本卓宏;田所勇樹;川上裕;Akira Sakai;A. Shioura;Minoru YAMAMOTO;田所勇樹;Yu Kawakami;山本卓宏;Akira Sakai;山本卓宏;Akira Sakai;田所勇樹;Akiyoshi Shioura;坂井哲;Yu Kawakami;山本卓宏;A. Shioura;川上裕;山本卓宏;坂井哲;田所勇樹;T.Yamamoto
• 通讯作者: T.Yamamoto

The minimal numbers of singularities of stable maps between surfaces

表面之间稳定映射的奇异点的最少数量

• DOI: 10.1016/j.topol.2009.06.010
• 发表时间: 2009
• 期刊: Topology and its Applications
• 影响因子: 0.6
• 作者: Akira Sakai;Lung-Chi Chen;山本稔;A.Kamenosono and T.Yamamoto
• 通讯作者: A.Kamenosono and T.Yamamoto

Geography of the singularities of stable maps of closed surfaces into the plane

封闭曲面稳定映射到平面的奇点的地理

• 发表时间: 2012
• 作者: Satoko Moriguchi;Akiyoshi Shioura;Nobuyuki Tsuchimura;T.Yamamoto
• 通讯作者: T.Yamamoto