Grassmann algebra with half infinite forms and its applications to the global analysis of mapping spaces

1.Halberd空间上微分运算的正则化考虑Hilbert空间H和H上的几类Schatten类算子G的配对，利用G的谱，提出了H上微分算子的正则化，并在适当的边界条件下编写了正则化的LoPlace算子和Dirac算子的适当的值和函数。利用这一结果，我们还阐明了Zeta正则化的含义以及Zeta正则化与H上的Cli Hord代数的无限旋量的关系。解决了Elizabeth alde提出的一个关于带质量项的Dirac算子的Zeta正则化行列式的问题，给出了Zeta正则化行列式的符号与质量项之间的关系。Cohen-Macauley代数的研究。Nisida研究了与本项目主题相关的代数。特别是在最小内射归结和链线方面做了很好的研究。关于可积方程的研究：Nakayama证明了AKNS逆散射格式的所有可积方程都是由曲线在Minkowski空间中的运动及其群论原因得到的。

1. Regularization of differential operation on a Halberd space.Considering the pair of a Hilbert space H and some Kinds of Schatten class operator G on H, we propose a regularization of differential operators on H by using spectuer of G. We compile proper values and functions of regularized Loplacian and Dirac operator with suitable boundary conditions. We also clarify the meaning of zeta regularization and the relation of zeta regularization and infinite spinor adgoisred to the Cli Hord algebra over H by using this result.2. Zeta regularized determinant of differential operators.We solved a problem presented by Elizalde on the zeta regularized determinant of Dirac operators with mass-terms and give the relation between the sign of zeta regularized determinant and mass-term.3. Study on Cohen-Macauley algebras.Nisida studied algebras related to the thema of this project. Especially on the minimal injective resolution and Catenary and well studied.4. Study on integrable equations.Nakayama showed all integrable equations by the AKNS inverse scattering schemer are obtained from the movement of curves in a hyperboloid in the Minkowski space and gene its group theoretical reason.