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These notes are a slightly enlarged version of my habilitation thesis, where our research interest and main results in the past few years are summarized. Most of the discussion revolves around complex ordinary differential equations and their underling foliations, singularity theory and dynamical systems. Compared to the original text, a section containing some background material on holomorphic foliations was added. Also some new results obtained in the past three years that are in line with the one presented in the habilitation were included.

In dimensions greater than or equal to 3, the local structure of a singular holomorphic foliation conceals a globally defined foliation on the projective space of dimension one less. In this paper, we will discuss how the global dynamics of the latter foliation exerts influence on several problems that apparently have a purely local nature. In the course of the discussion, a few recent results and open problems in the area will be reviewed as well.

This paper is devoted to geodesic completeness of left-invariant metrics for real and complex Lie groups. We start by establishing the Euler-Arnold formalism in the holomorphic setting. We study the real Lie group $\mathrm{SL}(2, \mathbb{R})$ and reobtain the known characterization of geodesic completeness and, in addition, present a detailed study where we investigate the maximum domain of definition of every single geodesic for every possible metric. We investigate completeness and semicompleteness of the complex geodesic flow for left-invariant holomorphic metrics and, in particular, establish a full classification for the Lie group $\mathrm{SL}(2, \mathbb{C})$., Comment: 34 pages, 18 figures

Let $\mathcal{F}$ be a foliation defined on a complex projective manifold $M$ of dimension $n$ and admitting a holomorphic vector field $X$ tangent to it along some non-empty Zariski-open set. In this paper we prove that if $X$ has sufficiently many integral curves that are given by meromorphic functions defined on $\mathbb{C}$ then the restriction of $\mathcal{F}$ to any invariant complex $2$-dimensional analytic set admits a first integral of Liouvillean type. In particular, on $\mathbb{C}^3$, every rational vector fields whose solutions are meromorphic functions defined on $\mathbb{C}$ admits a non-empty invariant analytic set of dimension $2$ where the restriction of the vector field yields a Liouvillean integrable foliation.

This paper is devoted to characterizing complex projective structures defined on Riemann surface orbifolds and giving rise to injective developing maps defined on the monodromy covering of the surface (orbifold) in question. The relevance of these structures stems from several problems involving vector fields with uniform solutions as well as from problems about "simultaneous uniformization" for leaves of foliations by Riemann surfaces. In this paper, we first describe the local structure of the mentioned projective structures showing, in particular, that they are locally bounded. In the case of Riemann surface orbifolds of finite type, the previous result will then allow us to provide a detailed global picture of these projective structures by exploiting their connection with the class of "bounded covering projective structures".

We provide examples of vector fields on $(\mathbb{C}^3, 0)$ admitting a formal first integral but no holomorphic first integral. These examples are related to a question raised by D. Cerveau and motivated by the celebrated theorems of Malgrange and Mattei-Moussu.

Starting from some remarkable singularities of holomorphic vector fields, we construct (open) complex surfaces over which the singularities in question are realized by complete vector fields. Our constructions lead to manifolds and vector fields beyond the algebraic setting and provide examples of complete vector fields with some new dynamical phenomena., Comment: to appear in Moscow Mathematical Journal

We classify degenerate singular points of $\C^2$-actions on complex surfaces.

This article presents the results of an experiment in which we investigated how prior algebra knowledge and personality can influence the permanence time from the confusion state to frustration/boredom state in a computer learning environment. Our experimental results indicate that people with a neurotic personality and a low level of algebra knowledge can deal with confusion for less time and can easily feel frustrated/bored when there is no intervention. Our analysis also suggest that people with an extroversion personality and a low level of algebra knowledge are able to control confusion for longer, leading to later interventions. These findings support that it is possible to detect emotions in a less invasive way and without the need of physiological sensors or complex algorithms. Furthermore, obtained median times can be incorporated into computational regulation models (e.g. adaptive interfaces) to regulate students' emotion during the teaching-learning process., Comment: 13 pages, 1 figure

This paper is devoted to the resolution of singularities of holomorphic vector fields and of one-dimensional holomorphic foliations in dimension 3 and it has two main objectives. First, from the general perspective of one-dimensional foliations, we build upon the work of Cano-Roche-Spivakovsky and essentially complete it. As a consequence, we obtain a general resolution theorem comparable to the resolution theorem of McQuillan-Panazzolo but proved by means of rather different methods. The second objective of this paper consists of looking at a special class of singularities of foliations containing, in particular, all singularities of complete holomorphic vector fields on complex manifolds of dimension 3. We then prove that for this class of holomorphic foliations, there holds a much sharper resolution theorem. This second result was the initial motivation of this paper and it relies on the combination of the previous resolution theorems for (general) foliations with some classical material on asymptotic expansions for solutions of differential equations., Comment: To appear in Russian Mathematical Surveys

We show that holomorphic vector fields on (C^3,0) have separatrices provided that they are embedded in a rank 2 representation of a two-dimensional Lie algebra. In turn, this result enables us to show that the second jet of a holomorphic vector field defined on a compact complex manifold M of dimension 3 cannot vanish at an isolated singular point provided that M carries more than a single holomorphic vector field.

In this note we extend to arbitrary dimensions a couple of results due respectively to Mattei-Moussu and to Camara-Scardua in dimension 2. We also provide examples of singular foliations having a Siegel-type singularity and answering in the negative the central question left open in the previous work of Camara-Scardua., Comment: This paper together with the paper "Discrete orbits, recurrence and solvable subgroups of Diff(C^2,0)" correspond to an expanded and accurate version of our previous preprint "Discrete orbits and special subgroups of Diff(C^n,0)" arXiv:1301.3716

We discuss the local dynamics of a subgroup of Diff(C^2,0) possessing locally discrete orbits as well as the structure of the recurrent set for more general groups. It is proved, in particular, that a subgroup of Diff(C^2,0) possessing locally discrete orbits must be virtually solvable. These results are of considerable interest in problems concerning integrable systems., Comment: The first version of this paper and "A note on integrability and finite orbits for subgroups of Diff(C^n,0)" are an expanded version of our paper "Discrete orbits and special subgroups of Diff(C^n,0)". An intermediate version re-submitted to the journal on March 2015 is available at http://www.fep.up.pt/docentes/hreis/publications.htm where there is also a comparison between these 3 versions

Consider a pseudogroup on (C,0) generated by two local diffeomorphisms having analytic conjugacy classes a priori fixed in Diff(C,0). We show that a generic pseudogroup as above is such that every point has (possibly trivial) cyclic stabilizer. It also follows that these generic groups possess infinitely many hyperbolic orbits. This result possesses several applications to the topology of leaves of foliations and we shall explicitly describe the case of nilpotent foliations associated to Arnold's singularities of type A^{2n+1}.

The local topological dynamics of subgroups of ${\rm Diff} ({\mathbb C^n}, 0)$, with special emphasis on ${\rm Diff} ({\mathbb C^2}, 0)$, is discussed with a view towards integrability questions. It is proved in particular that a subgroup of ${\rm Diff} ({\mathbb C^2}, 0)$ possessing locally finite orbits is necessarily solvable. Other results and examples related to higher-dimensional generalizations of Mattei-Moussu's celebrated topological characterization of integrability are also provided. These examples also settle a fundamental question raised by the previous work of Camara-Scardua.

In this paper it is shown that the existence of two independent holomorphic first integrals for foliations by curves on (C^3,0) is not a topological invariant. More precisely, we provide an example of two topologically equivalent foliations such that only one of them admits two independent holomorphic first integrals. The existence of invariant surfaces over which the induced foliation possesses infinitely many separatrices possibly constitutes the sole obstruction for the topological invariance of complete integrability and a characterization of foliations admitting this type of invariant surfaces is also given.

We show that generically a pseudogroup generated by holomorphic diffeomorphisms defined about $0 \in \mathbb{C}$ is free in the sense of pseudogroups even if the class of conjugacy of the generators is fixed. This result has a number of consequences on the topology of leaves for a (singular) holomorphic foliation defined on a neighborhood of an invariant curve. In particular in the classical and simplest case arising from local foliations possessing a unique separatrix that is given by a cusp of the form $\{y^2-x^{2k+1}=0\}$, our results allow us to settle the problem of showing that a generic foliation possesses only countably many non-simply connected leaves and that this countable set is, indeed, infinite.

We introduce a method to estimate the size of the domain of definition of the solutions of a meromorphic vector field on a neighborhood of its pole divisor. The corresponding techniques are, in a certain sense, quantitative versions of some well-known phenomena related to the presence of metrics with positive curvature. Several applications of these ideas are provided including a type of "confinement theorem" for solutions of complete polynomial vector fields on $\mathbb{C}^n$ and obstructions for certain (germs of) vector fields to be realized by a global holomorphic vector field on a compact K\"ahler manifold. As a complement a new approach to certain classical equations is proposed and detailed in the case of Halphen equations., Comment: This is a revised version of arXiv:1111.2365

These are slightly informal lecture notes intended for graduate students about the standard local theory of holomorphic foliations and vector fields. Though the material presented here is well-known some of the proofs differs slightly from the classical arguments., Comment: 115 pages, 9 figures