We consider the Teichmüller flow on the unit cotangent bundle of the moduli space of compact Riemann surfaces with punctures. We show that it is exponentially mixing for the Ratner class of observables. More generally, this result holds for the restriction of the Teichmüller flow to an arbitrary connected component of stratum. This result generalizes [AGY] which considered the case of strata of...

The paper dwells on the problems of stability of dynamical systems on a time scale. The paper is divided into the following sections: local existence and uniqueness, dynamic inequalities, existence of extremal solutions, comparison results, linear variation of parameters, nonlinear variation of parameters, global existence and stability, comparison theorems, stability criteria, etc.

There is a natural action of SL(2, R) on the moduli space of translation surfaces, and this yields an action of the unipotent subgroup U = 1 * 0 1. We classify the U-invariant ergodic measures on certain special submanifolds of the moduli space. (Each submanifold is the SL(2, R)-orbit of the set of branched covers of a fixed Veech surface.) For the U-action on these submanifolds, this is an ana...

Abstract Let F q \mathbb{F}_{q} be a finite field of order

In this survey paper, we discuss the classical Cassels’ theorem on existence of small-height zeros of quadratic forms over Q and its many extensions, to different fields and rings, as well as to more general situations, such as existence of totally isotropic small-height subspaces. We also discuss related recent results on effective structural theorems for quadratic spaces, as well as Cassels’-...

Motivation: it would be great if we could understand the p-adic variation of the Up-eigenvalues of modular forms as the weights and/or levels varied. (More generally, it would be really great if we could understand how the local p-adic Galois representations attached to automorphic forms behave – for example, even very weak “equidistribution” results here would presumably imply very strong auto...

We formulate and prove a finite version of Vinogradov’s bilinear sum inequality.We use it together with Ratner’s joinings theorems to prove that the Mobius function is disjoint from discrete horocycle flows on Γ\SL2(R).

Algorithmic meta-theorems are general algorithmic results applying to a whole range of problems, rather than just to a single problem alone. They often have a logical and a structural component, that is they are results of the form: every computational problem that can be formalised in a given logic L can be solved efficiently on every class C of structures satisfying certain conditions. This p...

The decade since the assembly of the human genome has witnessed dramatic advances in understanding the genetic differences that distinguish individual humans and that are responsible for specific traits. Genomewide association studies (GWAS) in humans have identified common germline, or inherited, DNA variants that are associated with various common human diseases, including diabetes, heart dis...