Non commutative geometry and moduli spaces of holomorphic curves

我们主要有两个结果。一个是关于Casson手柄上的模空间的构造，Casson手柄是非紧光滑的四个流形。一般而言，要在非紧空间上构造模理论，有两个步骤需要完成，其中一个是Fredholm论，另一个是横截性理论。对于前者，我们已经在卡森的把手上做了。在本研究期间，我们利用一种近似方法建立了关于Casson句柄的横截性理论。另一个结果是在二元多项式型偏微分方程组上建立了一个整体解析关系。它是通过不同偏微分方程解之间的粗略渐近比较而给出的。特别地，我们通过分析模空间关于这种关系的结构，证明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.