A uniform space is a topological space together with some additional structure which allows one to make sense of uniform properties such as completeness or uniform convergence. Motivated by previous work of J. Rivera-Letelier, we give a new construction of the Berkovich analytic space associated to an affinoid algebra as the completion of a canonical uniform structure on the associated rigid-an...

— We construct a rigid-analytic map from the the author’s half-integral weight cuspidal eigencurve (see [10]) to its integral weight counterpart that interpolates the classical Shimura lifting.

We prove two results about the local properties of generically one to one analytic mappings. Version: .76; Revised: 12–15–98; Run: March 23, 1999 §1: Introduction In this paper we consider two local properties of analytic maps which are generically one to one. First we consider holomorphic maps from open sets in C which behave locally like monoidal transformations: Let (z, w) denote coordinates...

This paper provides an overview of the theory of Bruhat-Tits buildings. Besides, we explain how Bruhat-Tits buildings can be realized inside Berkovich spaces. In this way, Berkovich analytic geometry can be used to compactify buildings. We discuss in detail the example of the special linear group. Moreover, we give an intrinsic description of Bruhat-Tits buildings in the framework of non-Archim...

Geometry over non–existent “field with one element” F1 conceived by Jacques Tits [Ti] half a century ago recently found an incarnation, in several related but different guises. In this paper I analyze the crucial role of roots of unity in this geometry and propose a version of the notion of “analytic functions” over F1. The paper combines a focused survey of various approaches with some new con...

Let V ⊂ R be a compact real analytic surface with isolated singularities, and assume its smooth part V0 is equipped with a Riemannian metric that is induced from some analytic Riemannian metric on R . We prove: 1. Each point of V has a neighborhood which is quasi-isometric (naturally and ’almost isometrically’) to a union of metric cones and horns, glued at their tips. 2. A full asymptotic expa...

We prove an index theorem concerning the pushforward of flat B-vector bundles, where B is an appropriate algebra. We construct an associated analytic torsion form T . If Z is a smooth closed aspherical manifold, we show that T gives invariants of π∗(Diff(Z)).

We study deformation theory for quantum integrable systems and prove several theorems concerning the Gevrey convergence and the unicity of perturbative expansions. À V.I. Arnold pour ses 70 ans.

V. L. Rvachev called R-functions “logically-charged functions” because they encode complete logical information within the standard setting of real analysis. He invented them in the 1960s as a means for unifying logic, geometry, and analysis within a common computational framework – in an effort to develop a new computationally effective language for modeling and solving boundary value problems...

Elsewhere in this issue Ferdinand Verhulst described the discussion of the interaction of analysis and geometry in the 19th century. In modern times such discussions come up again and again. As of 2014, synthetic geometry will not be part of the Dutch ‘vwo – mathematics B’ programme any more. Instead, the focus will be more on analytic geometry. Mark Timmer and Nellie Verhoef explored possibili...