Related papers: Markoff Triples and Strong Approximation

We show that turbulent dynamics that arise in simulations of the three-dimensional Navier--Stokes equations in a triply-periodic domain under sinusoidal forcing can be described as transient visits to the neighborhoods of unstable…

We study a family of birational maps of smooth affine quadric 3-folds, {over the complex numbers}, of the form $x_1x_4-x_2x_3=$ constant, which seems to have some (among many others) interesting/unexpected characters: a) they are…

The Medvedev degree of a subshift is a dynamical invariant of computable origin that can be used to compare the complexity of subshifts that contain only uncomputable configurations. We develop theory to describe how these degrees can be…

We study four distinct second-order nonlinear equations of Rabelo which describe pseudospherical surfaces. By transforming these equations to the constant-characteristic form we relate them to some well-studied integrable equations. Two of…

Given an affine variety $X$, a morphism $\phi:X\to X$, a point $\alpha\in X$, and a Zariski closed subset $V$ of $X$, we show that the forward $\phi$-orbit of $\alpha$ meets $V$ in at most finitely many infinite arithmetic progressions, and…

Using valuative techniques, we show that a smooth affine surface with a non-elementary automorphism group and completable by a cycle of rational curves is either the algebraic torus or a smooth cubic affine surface of Markov type.…

We study the exponential map of connected symmetric spaces and characterize, in terms of midpoints and of infinitesimal conditions, when it is a diffeomorphism, generalizing the Dixmier-Saito theorem for solvable Lie groups. We then give a…

Given a finite residue field $k$, one looks for a smoothness basis that is invariant under the automorphism group of $k$. We construct models for some finite fields that admit such a basis. This work aims at accelerating algorithms for…

It has long been known that to a complex cubic surface or threefold one can canonically associate a principally polarized abelian variety. We give a construction which works for cubics over an arithmetic base. This answers, away from the…

Let $X$ be a smooth algebraic variety endowed with an action of a finite group $G$ such that there exists the geometric quotient $\pi_X:X\to X/G$. We characterize rational tensor fields $\tau$ on $X/G$ such that the {\it pull back} of $\tau…

This is a survey on the transitive quantum groups $G\subset S_N^+$, and on the flat matrix models $\pi:C(G)\to M_N(C(X))$ for the corresponding Hopf algebras. We review the known results on the subject, with a number of improvements,…

We prove new cases of Vojta's conjectures for surfaces in the context of function fields, with truncation equal to one and providing an effective explicit description of the exceptional set. We also prove a general and explicit result…

We initiate the study of random iteration of automorphisms of real and complex projective surfaces, or more generally compact K{\"a}hler surfaces, focusing on the fundamental problem of classification of stationary measures. We show that,…

The main result of the present paper is a construction of relative moduli spaces of stable sheaves over the stack of quasipolarized projective surfaces. For this, we use the theory of good moduli spaces, whose study was initiated by Alper.…

On a polarised surface, solutions of the Vafa-Witten equations correspond to certain polystable Higgs pairs. When stability and semistability coincide, the moduli space admits a symmetric obstruction theory and a $\mathbb C^*$ action with…

We classify the affine connections on compact orientable surfaces for which the pseudogroup of local isometries acts transitively. We prove that such a connection is either torsion-free and flat, the Levi-Civita connection of a Riemannian…

We extend some parts of the representation theory for integral quadratic forms over the ring of integers of a number field to the case over the coordinate ring $k[C]$ of an affine curve $C$ over a general base field $k$. By using the genus…

The paper considers a system of equations that models a lateral flow of a Boussinesq--Scriven fluid on a passively evolving surface embedded in $\mathbb{R}^3$. For the resulting Navier-Stokes type system, posed on a smooth closed…

The main purpose of this paper is to show that Markov solutions to the 3D Navier--Stokes equations driven by Gaussian noise have the strong Feller property up to the critical topology given by the domain of the Stokes operator to the power…

We say that a group G acts infinitely transitively on a set X if for every integer m the induced diagonal action of G is transitive on the cartesian mth power of X with the diagonals removed. We describe three classes of affine algebraic…