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

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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.