This is a survey on Okutsu-Montes representations of prime ideals of certain one-dimensional integral closures. These representa- tions facilitate the computational resolution of several arithmetic tasks concerning prime ideals of global fields.

We study some numerical properties of singularities of codimension one holomorphic foliations which can be analytically collapsed to one point. Some local and global dynamical consequences are deduced.

We study, in $L^{1}({\mathbb R}^n;\gamma)$ with respect to the gaussian measure, non-tangential maximal functions and conical square functions associated with the Ornstein-Uhlenbeck operator by developing a set of techniques which allow us, to some extent, to compensate for the non-doubling character of the gaussian measure. The main result asserts that conical square functions can be controlled in $L^1$-norm by non-tangential maximal functions. Along the way we prove a change of aperture result for the latter. This complements recent results on gaussian Hardy spaces due to Mauceri and Meda.

Every two-variable rational inner function on the bidisk has a special representation called a unitary transfer function realization. It is well known and related to important ideas in operator theory that this does not extend to three or more variables on the polydisk. We study the class of rational inner functions on the polydisk which do possess a unitary realization (the Schur-Agler class) and investigate minimality in their representations. Schur-Agler class rational inner functions in three or more variables cannot be represented in a way that is as minimal as two variables might suggest.

We study spaces of realisations of linkages (weighted graphs) whose underlying graph is a series parallel graph. In particular, we describe an algorithm for determining whether or not such spaces are connected.

Let $\operatorname{Fol}_{\mathbb{R}}(2,d)$ be the space of real algebraic foliations of degree $d$ in $\mathbb{R} \mathbb{P}(2)$. For fixed $d$, let $\operatorname{Int}_{\mathbb{R}}(2,d)=\lbrace\mathcal{F}\in \operatorname{Fol}_{\mathbb{R}}(2,d)\mid \mathcal{F}$ has a non-constant rational first integral$\rbrace$. Given $\mathcal{F}\in \operatorname{Int}_\mathbb{R}(2,d)$, with primitive first integral~$G$, set $O(\mathcal{F})=$ number of real ovals of the generic level $(G=c)$. Let $O(d)=\sup\lbrace O(\mathcal{F})\mid \mathcal{F}\in \operatorname{Int}_{\mathbb{R}}(2,d)\rbrace$. The main purpose of this paper is to prove that $O(d)=+\infty$ for all $d\ge5$.

We prove the existence of regular foliations with a prescribed tangency divisor in neighborhoods of negatively embedded holomorphic curves; this is related to a linearization theorem due to Grauert. We give also examples of neighborhoods which can not be linearized.

We define metrics on Culler-Vogtmann space, which are an analogue of the Thurston metric and are constructed using stretching factors. In fact the metrics we study are related, one being a symmetrised version of the other. We investigate the basic properties of these metrics, showing the advantages and pathologies of both choices.

We show how to compute stretching factors between marked metric graphs in an easy way and we discuss the behaviour of stretching factors under iterations of automorphisms.

We study metric properties of folding paths, showing that they are geodesic for the non-symmetric metric and, if they do not enter the thin part of Outer Space, quasi-geodesic for the symmetric metric.

Using Wilson's Haar basis in $\mathbb{R}^n$, which is different than the usual tensor product Haar functions, we define its associated dyadic paraproduct in $\mathbb{R}^n$. We can then extend "trivially'' Beznosova's Bellman function proof of the linear bound in $L^2(w)$ with respect to $[w]_{A_2}$ for the 1-dimensional dyadic paraproduct. Here trivial means that each piece of the argument that had a Bellman function proof has an $n$-dimensional counterpart that holds with the same Bellman function. The lemma that allows for this painless extension we call the good Bellman function Lemma. Furthermore the argument allows to obtain dimensionless bounds in the anisotropic case.

In this paper we have obtained the boundedness of bilinear Littlewood-Paley operators on the circle group ${\mathbb T}$ by using appropriate transference techniques. In particular, bilinear analogue of Carleson's Littlewood-Paley result for all possible indices has been obtained. Also, we prove some bilinear analogues of de Leeuw's results concerning multipliers of ${\mathbb R}^n$.