This is a report on some joint work with Mark Reeder and Jiu-Kang Yu. I will review the theory of parahoric subgroups and consider the induced representation of a one-dimensional character of the pro-unipotent radical. A surprising fact is that this induced representation can (in certain situations) have finite length. I will describe the parahorics and characters for which this occurs, and what the Langlands parameters of the corresponding irreducible summands must be.

We study families of filtered phi-modules associated to families of p-adic Galois representations as considered by Berger and Colmez. We show that the weakly admissible locus in a family of filtered phi-modules is open and that the groupoid of weakly admissible modules is in fact an Artin stack. Working in the category of adic spaces instead of the category of rigid analytic spaces one can show that there is an open substack of the weakly admissible locus over which the filtered phi-modules is induced from a family of crystalline representations.

A "proof assistant" is a software package comprising a validity checker for proofs in a particular logic, accompanied by semi-decision procedures called "tactics" that assist the mathematician in filling in the easy parts of the proofs. I will demonstrate the use of the Coq proof assistant in doing simple proofs about inductive structures such as natural numbers, sequences, and trees.

The classical Dvoretzky theorem asserts that for every integer k>1 and every target distortion D>1 there exists an integer n=n(k,D) such that any