Related papers: Difference fields and descent in algebraic dynamic…

We establish the dynamical Northcott property for polarized endomorphisms of a projective variety over a function field $\mathbf{K}$ of characteristic zero, and we relate this property to the notion of stability in complex dynamics. This…

This paper concerns the model theory of jet spaces (i.e., higher-order tangent spaces) in differentially closed fields. Suppose p is the generic type of the jet space to a finite dimensional differential-algebraic variety at a generic…

In this paper a geometric field theory of dislocation dynamics and finite plasticity in single crystals is formulated. Starting from the multiplicative decomposition of the deformation gradient into elastic and plastic parts, we use…

Discussed is relationship between nonlinearity and symmetry of dynamical models. The special stress is laid on essential, non-perturbative nonlinearity, when none linear background does exist. This is nonlinearity essentially different from…

We characterize universally generalizing morphisms which satisfy descent of algebraic cycles integrally as those universally generalizing morphisms which are surjective with generically reduced fibres. In doing so, we introduce a naive…

We establish a structure theorem for the integral points on moduli of special linear rank two local systems over surfaces, using mapping class group descent and boundedness results for systoles of local systems.

This paper develops moving frame theory for partial difference equations and for differential-difference equations with one continuous independent variable. In each case, the theory is applied to the invariant calculus of variations and the…

In this paper, we study arithmetic dynamics in arbitrary characteristic, in particular in positive characteristic. We generalise some basic facts on arithmetic degree and canonical height in positive characteristic. As applications, we…

Let C be an algebraically closed field and X a projective curve over C. Consider an ordinary linear differential equation, or a linear differ- ence equation, with coefficients in the field of rational functions of X, and assume that its…

In this thesis three topics on the model theory of partial differential fields are considered: the generalized Galois theory for partial differential fields, geometric axioms for the theory of partial differentially closed fields, and the…

I analyse a generalised Burger's equation to develop an accurate finite difference approximation to its dynamics. The analysis is based upon centre manifold theory so we are assured that the finite difference model accurately models the…

Given a diagram of schemes, we can ask if a geometric object over one of them can be built from descent data (usually objects of the same type over the various other schemes in the diagram, together with compatibility isomorphisms). Using…

We formulate a refined theory of linear systems, using the methods of a previous paper, "A Theory of Branches for Algebraic Curves", and use it to give a geometric interpretation of the genus of an algebraic curve. Using principles of…

The generalization of the n-dimensional cube, an n-dimensional chain, the exterior derivative and the integral of a differential n-form on it are introduced and investigated. The analogue of Stokes theorem for the differential space is…

A new tool for the model theory of differentially closed fields and of compact complex manifolds is here developed. In such settings, it is shown that a type internal to the field of constants (resp. to the projective line) admits a maximal…

The descent algebra of the symmetric group, over a field of non-zero characteristic p, is studied. A homomorphism into the algebra of generalised p-modular characters of the symmetric group is defined. This is then used to determine the…

In one of the fundamental results of Arakelov's arithmetic intersection theory, Faltings and Hriljac (independently) proved the Hodge Index Theorem for arithmetic surfaces by relating the intersection pairing to the negative of the…

This article studies descent theory in the setting of Berkovich spaces. We give sufficient conditions for a given fibered category over the category of k-affinoid algebras to be a stack for the Berkovich analogue of the faithfully-flat…

We present a differential algebra of generalized functions over a field of generalized scalars by means of several axioms in terms of general algebra and topology. Our differential algebra is of Colombeau type in the sense that it contains…

Motivated by applications in moduli theory, we introduce a flexible and powerful language for expressing lower bounds on relative dimension of morphisms of schemes, and more generally of algebraic stacks. We show that the theory is robust…