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Topology related to Mathematical Physics, Morse Theory and Numerical Computations

与数学物理、莫尔斯理论和数值计算相关的拓扑

基本信息

批准号：
16540056
负责人：
YAMAGUCHI Kohhei
金额：
$ 2.18万
依托单位：
The University of Electro-Communications
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540056/
关键词：
projective variety polynomial holomorphic map homotopy type fundamental group real projective space algebraic geometry map ホモロジー群複素射影空間ポアンカレ複体多様体ホップ写像代数写像 labeled configuration space truncated configuration space m-twi

项目摘要

Previously, Professors M.Guest, A.Kozlowski and the author showed that the Atiyah-Jones-Segal type Theorem holds for spaces of holomorphic maps from the 1 dimensional complex projective space to certain family of complex projective varieties. Now he showed that a similar result holds for certain subspaces of them which are defined by using the concept of multiplicities induced from the representations of polynomials of holomorphic maps. Furthermore, he computed the fundamental groups for spaces of self-holomorphic maps on the n dimensional complex Projective spaces.Until now, we usually investigate whether AJS type Theorem holds or not for spaces of holomorphic (or algebraic) maps from one real dimensional (or complex one dimensional) spaces. In our investigation, now we can investigate whether such a problem for spaces of holomorphic or algebraic maps from high dimensional spaces. As one example, we can show that the spaces of regular maps from certain compact affine spaces into complex or real Grassmanian manifolds are homotopy equivalent of spaces of continuous maps between these spaces if these varieties Affine spaces satisfy certain conditions of vector bundles, which is one of joint works with Professor A. Kozlowski. To prove these results, we use the technique of real algebraic geometry. Moreover, we can prove that AJS type Theorem holds for such spaces by using the above Theorem. In particular, we also determine the fundamental groups of spaces of maps from m dimensional real projective space into n dimensional one when m=n-1, or m=n. Such a result can be regarded as a real version of the study investigated in the above first case.We also study the exceptional surgery from the new point view of singularity theory by using the divide theory. In particular, we study the mechanism of such surgeries and the structure of the set of exceptional surgeries.

在此之前，M.Guest，A.Kozlowski教授和作者证明了Atiyah-Jones-Segal型定理对于从1维复射影空间到某一族复射影体族的全纯映射空间成立。现在，他证明了类似的结果也适用于它们的某些子空间，这些子空间是由全纯映射的多项式的表示导出的重数的概念定义的。此外，他还计算了n维复射影空间上自全纯映射空间的基本群。直到现在，我们通常都在研究AJS型定理对于一维(或一维复数)空间的全纯(或代数)映射空间是否成立。在我们的研究中，我们现在可以研究高维空间中的全纯映射或代数映射空间是否存在这样的问题。作为一个例子，我们可以证明从某些紧仿射空间到复或实Grassmanian流形的正则映射空间是这些空间之间的连续映射空间的同伦等价，如果这些簇仿射空间满足向量丛的某些条件，这是A.Kozlowski教授的共同工作之一。为了证明这些结果，我们使用了实代数几何的技巧。此外，我们还利用上述定理证明了AJS型定理在此类空间中成立。特别地，当m=n-1或m=n时，我们还确定了从m维实射影空间到n维实射影空间的映射的基本群.这样的结果可以看作上述第一种情况下研究的实数版.我们还利用除法理论从奇点理论的新角度研究了例外运算.特别是，我们研究了这类手术的机制和例外手术集的结构。

项目成果

期刊论文数量（74）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

On periodicizing functions

关于周期化函数

DOI：
发表时间：
2006
期刊：
Bull. Korean Math. Soc. 43・2
影响因子：
0
作者：
T.Naito;S.J.Son
通讯作者：
S.J.Son

The homotopy of spaces of maps between real projective spaces

实射影空间之间的映射空间的同伦

DOI：
发表时间：
2006
期刊：
J. Math. Soc. Japan 58-4
影响因子：
0
作者：
小島定吉;糸川銚;酒井隆;戸田正人;小林治;二木昭人;浦川肇;十文字正樹;山口孝男;塩谷隆;小林亮一;T.Sakai;T. Sakai;Shingo Okuyama;Kazuhisa Shimakawa;Kohhei Yamaguchi
通讯作者：
Kohhei Yamaguchi

New characterizations of exponential dichotomy and exponential stability of linear differential equations

线性微分方程指数二分法和指数稳定性的新表征