Non commutative geometry and moduli spaces of holomorphic curves

主要有两个结果。一个是关于非紧光滑四流形卡森柄上模空间的构造。一般来说，为了在非紧空间上构造模理论，有两个步骤需要完成，一个是Fredholm理论，另一个是横向理论。对于前者，我们已经完成了卡森手柄。在此研究期间，我们利用近似方法构建了卡森手柄的横向理论。另一个结果是在两变量多项式型偏微分方程的集合上构造了一个全局解析关系。利用不同偏微分方程正解之间的粗糙渐近比较给出了它。特别地，我们通过分析模空间相对于这种关系的结构，证实了这种关系是非平凡的。

Mainly we have two results. One is on the construction of moduli spaces over Casson handles which are non compact smooth four manifolds. In general in order to construct moduli theory over non compact spaces, there are two steps which should be done, where one is Fredholm theory, and the other is transversality theory. For the former we have already done over the Casson handles. During this research periods, we have constructed transversality theory over the Casson handles by use of an approximation method. The other result is to construct a globally analytic relation on the set of polynomial type partial differential equations of two variables. It is given by use of a rough asympototic comparison between positive solutions to different PDEs. In particular we have verified that the relation is non trivial by analyzing the structure of the moduli space with respect to the relation.