超级分析讲座--为什么需要和那是什么？
Lectures on Super Analysis -- Why necessary and What's that?
摘要：
粗略地说，RA（=实分析）意味着研究定义在实空间上的（光滑）函数的性质，CA（=复分析）代表研究定义在具有复结构的空间上的（全纯）函数的性质。但要平等对待玻色子和费米子，我们需要准备一个“基环”，Fréchet-Grassmann代数有可数多个Grassmann生成元，我们定义了所谓的超空间在这样的代数。在这样的超空间上，我们引入了超光滑函数空间，并发展了初等微积分。稍微准备功能分析，我们解释Efetov的方法在RMT（=随机矩阵理论）。自由Weyl方程被用来回答Feynman在他们著名的著作中提出的问题。简单的例子SUSYQM（=超对称量子力学）计算从这个角度来看。最后一章给出了Berezin整号变元公式的精确证明，超空间实分析的简要介绍，齐氏弱双曲方程基本解的另一种构造，Bernardi关于Egorov定理的系统形式的问题等，并对作为数学新分支候选者的泛函微分方程作了一些讨论。最后，我们构造了一个对应于具有外电磁势的Weyl方程的Hamilton流，其中我们需要可数无限Grassmann生成元和弱拓扑！我提到了许多悬而未决的问题，至少对我来说是这样（别名ATLOM=一个小小的老数学家）。
英文摘要：
Roughly speaking, RA(=real analysis) means to study properties of (smooth) functions defined on real space, and CA(=complex analysis) stands for studying properties of (holomorphic) functions defined on spaces with complex structure. But to treat boson and fermion on equal footing, we need to prepare as a "ground ring", Fréchet-Grassmann algebra having countably many Grassmann generators and we define so-called superspace over such algebra. On such superspaces, we introduce spaces of super-smooth functions and develop elementary differential and integral calculus. With a slight preparation of functional analysis, we explain the Efetov's method in RMT(=random matrix theory). The free Weyl equation is treated to answer the problem posed by Feynman in their famous book. Simple examples of SUSYQM(=supersymmetric quantum mechanics) are calculated from this point of view. In the final chapter, we give a precise proof of Berezin's formula for changing variables under integral sign, brief introduction of real analysis on superspace, another construction of fundamental solution of Qi's weakly hyperbolic equation, Bernardi's question for a system version of Egorov's theorem, etc. A little discussion of Funtional Derivative Equations which are candidates of new branch of mathematics, is given. And finally, we construct a Hamilton flow corresponding to the Weyl equation with external electro-magnetic potentials, where we need the countably infinite Grassmann generators and weak topology! I mention many open problems at least for me (alias ATLOM=a tiny little old mathematician).
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