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

• Applicant: Professor Dr. Helge Glöckner
• Subject Area: Mathematics
• Term: from 2009 to 2012
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 118701543

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.

• DFG Programme: Research Grants