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In this thesis we present the results of four years of research on some aspects of p-adic geometry. A first result is on definable Lipschitz extensions of p-adic functions, a second result is on differentiation in P-minimal structures. Both results will appear in the form of an article in a peer-reviewed mathematical journal.Firstly, we prove a definable version of Kirszbrauns theorem in a non-Archimedean setting for definable families of functions in one variable. More precisely, let K be a finite field extension of the field of p-adic numbers, then we prove that every definable family of λ-Lipschitz functions on a subset of K extends to a definable family of λ-Lipschitz functions on K.Secondly, we prove a p-adic, local version of the Monotonicity Theorem for P-minimal structures. The existence of such a theorem was originally conjectured by Haskell and Macpherson. We approach the problem by considering the first order strict derivative. In particular, we show that, for a wide class of P-minimal structures, the definable functions are almost everywhere strictly differentiable and satisfy the Local Jacobian Property.The basic facts of p-adic fields are reviewed in the first chapter of this thesis. Model theory and its applications are reviewed in the second chapter. The third chapter contains the new results on definable Lipschitz extensions of p-adic functions, and the forth chapter those on differentiation in P-minimal structures. Finally, the fifth chapter contains a discussion of the main results in this thesis, together with a look at future research.

512.7 <043.3> --- Academic collection --- Algebraic geometry. Commutative rings and algebras--dissertaties --- Theses