Related papers: Heights and arithmetic dynamics over finitely gene…

Let R be a commutative, noetherian, local ring. Topological Q-vector spaces modelled on full subcategories of the derived category of R are constructed in order to study intersection multiplicities.

We establish the finiteness of periodic points, that we called Geometric Dynamical Northcott Property, for regular polynomials automorphisms of the affine plane over a function field $\mathbf{K}$ of characteristic zero, improving results of…

Vector fields with components which are generalized zero-forms are constructed. Inner products with generalized forms, Lie derivatives and Lie brackets are computed. The results are shown to generalize previously reported results for…

Let $K$ be an algebraically closed field of arbitrary characteristic, $X$ an irreducible variety and $Y$ an irreducible projective variety over $K$, both are not necessarily smooth. Let $f:X\rightarrow X$ and $g:Y\rightarrow Y$ be dominant…

On complex algebraic varieties, height functions arising in combinatorial applications fail to be proper. This complicates the description and computation via Morse theory of key topological invariants. Here we establish checkable…

For a smooth, projective complex variety, we introduce several mixed Hodge structures associated to higher algebraic cycles. Most notably, we introduce a mixed Hodge structure for a pair of higher cycles which are in the refined normalized…

Multivector fields and differential forms at the continuum level have respectively two commutative associative products, a third composition product between them and various operators like $\partial$, $d$ and $*$ which are used to describe…

We investigate correspondence functors, namely the functors from the category of finite sets and correspondences to the category of $k$-modules, where $k$ is a commutative ring.They have various specific properties which do not hold for…

We use height arguments to prove two results about the dynamical Mordell-Lang problem. (i) For an endomorphism of a projective variety, the return set of a dense orbit into a curve is finite if any cohomological Lyapunov multiplier of any…

A model of K-homology with coefficients in a mapping cone using the framework of the geometric cycles of Baum and Douglas is developed. In particular, this leads to a geometric realization of K-homology with coefficients in R/Z. In turn,…

The Hodge equations for 1-forms are studied on Beltrami's projective disc model for hyperbolic space. Ideal points lying beyond projective infinity arise naturally in both the geometric and analytic arguments. An existence theorem for…

In this article, we generalize the arithmetic degree and its related theory to dynamical systems defined over an arbitrary field $\mathbf{k}$ of characteristic $0$. We first consider a dynamical system $(X,f)$ over a finitely generated…

We survey variety theory for modules of finite dimensional Hopf algebras, recalling some definitions and basic properties of support and rank varieties where they are known. We focus specifically on properties known for classes of examples…

We develop a sequential-topological study of rational points of schemes of finite type over local rings typical in higher dimensional number theory and algebraic geometry. These rings are certain types of multidimensional complete fields…

We introduce the concept of geometric extremal graphical models, which are defined through the gauge function of the limit set obtained from suitably scaled random vectors in light-tailed margins. For block graphs, we prove results relating…

We define an "ample canonical height" for an endomorphism on a projective variety, which is essentially a generalization of the canonical heights for polarized endomorphisms introduced by Call--Silverman. We formulate a dynamical analogue…

A multiscale theory of interacting continuum mechanics and thermodynamics of mixtures of fluids, electrodynamics, polarization and magnetization is proposed. The mechanical (reversible) part of the theory is constructed in a purely…

Roth's theorem is extended to finitely generated field extensions of $\Bbb Q$, using Moriwaki's framework for heights.

Let A, B, S be categories, let F:A-->S and G:B-->S be functors. We assume that for "many" objects a in A, there exists an object b in B such that F(a) is isomorphic to G(b). We establish a general framework under which it is possible to…

The concept of hypocoercivity for linear evolution equations with dissipation is discussed and equivalent characterizations that were developed for the finite-dimensional case are extended to separable Hilbert spaces. Using the concept of a…