Aspects of non-archimedian non-linear analysis and functional analysis

该项目旨在将实微积分、非线性分析和非线性泛函分析的思想转化为非阿基米德分析，即对超度域(如p-ady数)的分析。它致力于进一步发展超度场上的有限和无限维微积分，并将其应用于两个主要领域，即超度场上的动力系统和李理论。在这两个领域，映射和流形都是自然遇到的，它们不是解析的，而仅仅是光滑的、连续可微的或Hölder的。虽然解析结构在非阿基米德分析(特别是严格解析几何)中得到了深入的研究，但到目前为止，实微分的类比还没有引起太多的关注，至少在高维或无限维上是这样。对于超度量微积分，我们将证明Nash-Moser逆函数定理的一个p-进类，以及函数空间的新的指数律。在李理论中，目标是在超度场上构造新的无限维李群。关于动力系统的主题包括围绕超度场上流形的微分同胚的不动点构造不变叶。

The project aims to transfer ideas from real differential calculus, non-linear analysis and non-linear functional analysis into non-archimedean analysis, i.e., analysis over ultrametric fields (such as the p-adic numbers). It strives to develop further the finite- and infinitedimensional differential calculus over ultrametric fields and to apply it in two main areas, namely dynamical systems and Lie theory over such fields. In both areas, mappings and manifolds are encountered naturally which are not analytic, but merely smooth, continuously differentiable, or Hölder. While analytic structures have been studied intensively in non-archimedean analysis (notably in rigid analytic geometry), the analogue of real differential calculus has attracted less attention so far, at least in higher or infinite dimensions. As to ultrametric calculus, we intend to prove a p-adic analogue of the Nash-Moser inverse function theorem, as well as new exponential laws for function spaces. In Lie theory, the goals are to construct new classes of infinite-dimensional Lie groups over ultrametric fields. Topics concerning dynamical systems include the construction of invariant foliations around fixed points of diffeomorphisms of manifolds over ultrametric fields.