Topology related to Mathematical Physics, Morse Theory and Numerical Computations

先前，M.Guest， A.Kozlowski教授和作者证明了Atiyah-Jones-Segal型定理适用于从一维复射影空间到某一族复射影变体的全纯映射空间。现在他证明了一个类似的结果适用于它们的某些子空间，这些子空间是用由全纯映射的多项式表示导出的多重概念来定义的。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。到目前为止，我们通常研究AJS型定理是否适用于一个实维（或复一维）空间的全纯（或代数）映射空间。在我们的研究中，现在我们可以研究这个问题是否适用于高维空间的全纯或代数映射空间。作为一个例子，我们证明了从某些紧仿射空间到复或实格拉斯曼流形的正则映射的空间是这些空间之间的连续映射的空间的同伦等价，如果这些变体仿射空间满足向量束的某些条件，这是与A. Kozlowski教授的联合工作之一。为了证明这些结果，我们使用了实代数几何的技巧。此外，我们可以利用上述定理证明AJS型定理对这样的空间成立。特别地，我们还确定了当m=n-1或m=n时，从m维实射影空间映射到n维实射影空间的基本群。这样的结果可以被视为上述第一个案例中所调查的研究的真实版本。我们还从奇点理论的新观点出发，利用分割理论对异常手术进行了研究。特别地，我们研究了这种手术的机制和特殊手术集的结构。

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.