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

1. Halbert空间上微分算子的正则化考虑Hilbert空间H和H上的几类Schatten类算子G的对偶，利用G的谱给出了H上微分算子的正则化.在适当的边界条件下，我们编制了正则化的拉普拉斯算子和狄拉克算子的适当值和函数。利用这一结果，我们还阐明了zeta正则化的意义以及zeta正则化和无穷旋量adgoisred与H上Cli-Hord代数的关系.微分算子的Zeta正则化行列式，解决了Elizalde提出的关于Dirac算子的Zeta正则化行列式的一个问题，给出了Zeta正则化行列式的符号与质量项的关系。Cohen-Macauley代数的研究。Nisida研究了与本项目主题相关的代数。特别是对极小内射分解和悬链线进行了深入的研究.对可积方程的研究，Nakayama证明了AKNS逆散射格式下的所有可积方程都是由Minkowski空间中双曲面上的曲线运动得到的，并给出了其群论原因。

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.