Research on invariants of real analytic singularities

本研究的目的是寻找和引入实解析奇点在拓扑等价、爆破解析等价、Lipschitz等价、C^1等价等方面的不变量，并利用不变量和平凡定理给出关于这些等价的实解析奇点的分类。关于这些问题，我得到了如下结果：1)在与Adam Parusinski的合作中，我们引入了Motivic-型不变量作为实解析函数芽的爆破解析不变量，并利用我们的不变量和Fukui不变量给出了实解析函数的爆破解析分类。2)我与A.Parusinski一起研究了与上面不同的二元实解析函数的类型不变量。我们证明了关于圆弧的牛顿边界是Lipschitz不变量，带点的关于圆弧的牛顿边界在两个变量的情况下是C^1不变量更多的是实解析函数在芽中的吹解析不变量。另一方面，对于复全纯函数芽，Fukui不变量是否为拓扑不变量也是一个问题。4)对于半代数集之间的半代数映射，我和Mahiro Shiota研究了纤维不光滑的点集。特别是，我们为纤维光滑部分的平凡部分寻找了一个不变量。5)关于代数奇异族的Blow-Nash模问题，我与T.Fukui和K.Bekka一起证明了任何多项式映射都可以表示为某个代数簇的去奇异映射对该簇的严格变换和例外集在某一点的交集的限制。6)我引入了Lipschitz弧的海缠邻域的概念证明了邻域的度和宽度是biLipschitz不变量，并且利用不变量证明了Briancon-Speder族和Oka族不是biLpisitchz平凡的。此外，发展了这一思想，我与Laurentiu Paunescu得到了一个一般性的结果，即次解析集的亲吻维度是双Lipschitz不变量。较少

The aim of this research is to look for and introduce some invariants of real analytic singularities on topological equivalence, blow-analytic equivalence, Lipschitz equivalence, C^1 equivalence and so on, and to give classifications of real analytic singularities on those equivalences using the invariants and triviality theorems. Concerning these problems, I have obtained the following :1)In the joint work with Adam Parusinski, we have introduced motivic-type invariants as blow-analytic invariants for real analytic function germs, and have given blow-analytic classification of real analytic functions using our invariants and the Fukui invariants.2)I have worked with A.Parusinski also on a different type invariant from the above for real analytic functions of 2 variables. We have shown that the Newton boundary relative to an arc is a Lipschitz invariant and the Newton boundary with dots relative to an arc is a C^1 invariant in the 2 variables case.3)The Fukui invariant is well-known as … More a blow-analytic invariant for real analytic functioin germs. On the other hand, there is a problem whether the Fukui invariant is a topological invariant for complex holomorphic function germs. I have given the affirmative answer to it in the 2 variables case, and have constructed a negative example of 4 variables functions.4)For a semialgebraic mapping between semialgebraic sets, I and Mahiro Shiota have studied the set of points at which the fibre is not smooth. In particular, we have looked for an invariant for the triviality along the smooth part of a fibre. Then we have proved that a semialgebraic function is semialgebraically trivial along the smooth fibre, and constructed an example of a semialgebraic mapping which is not trivial along it.5)Related to the problem on Blow-Nash moduli for a family of algebraic singularities, I have proved with T.Fukui and K.Bekka that any polynomial mapping can be represented as the restriction of a desingularisation map of some algebraic variety to the intersection of the strict transform of the variety and the exceptional set at some point.6)I have introduced the notion of a sea-tangle neighborhood for a Lipschitz arc, and have shown that the degree and the width of the neighborhood are biLipschitz invariants and that the Briancon-Speder family and the Oka family are not biLpischitz trivial using the invariants. In addition, developing the idea, I have obtained a general result with Laurentiu Paunescu that the kissing dimension of subanalytic sets is a biLipschitz invariant. Less