Abstract:
in this work are studied periodic perturbations, depending on two parameters, of quadratic planar polynomial vector fields having an infinite heteroclinic cycle, which is an unbounded solution joining two saddle points at infinity. the global study envolving infinity is performed via the poincaré compactification. the main result obtained states that for certain types of periodic perturbations, the perturbed system has quadratic heteroclinic tangencies and transverse intersections between the local stable and unstable manifolds of the hyperbolic periodic orbits at infinity. it implies, via the birkhoff-smale theorem, in a complex dynamical behavior of the solutions of the perturbed system, in a finite part of the phase plane.

Abstract:
In this work are studied periodic perturbations, depending on two parameters, of quadratic planar polynomial vector fields having an infinite heteroclinic cycle, which is an unbounded solution joining two saddle points at infinity. The global study envolving infinity is performed via the Poincaré compactification. The main result obtained states that for certain types of periodic perturbations, the perturbed system has quadratic heteroclinic tangencies and transverse intersections between the local stable and unstable manifolds of the hyperbolic periodic orbits at infinity. It implies, via the Birkhoff-Smale Theorem, in a complex dynamical behavior of the solutions of the perturbed system, in a finite part of the phase plane.

Abstract:
This paper is complementary to the work Rosen-Silverman, which derives a criteria on the number fields for the independence of Heegner points associated to them on non-CM elliptic curves. We show that the same criteria holds for CM elliptic curves. A generalisation of the independence criteria for the Heegner points associated to an order of fixed conductor of quadratic imaginary fields can also be found in this paper.

Abstract:
We construct a polynomial planar vector field of degree two with one invariant algebraic curves of large degree. We exhibit an explicit quadratic vector fields which invariant curves of degree nine, twelve, fifteen and eighteen degree.

Abstract:
We consider a class of foliations on the complex projective plane that are determined by a quadratic vector field in a fixed affine neighborhood. Such foliations, as a rule, have an invariant line at infinity. Two foliations with singularities on $\mathbb C P^2$ are topologically equivalent provided that there exists a homeomorphism of the projective plane onto itself that preserves orientation both on the leaves and in $\mathbb C P^2$ and brings the leaves of the first foliation to that of the second one. We prove that a generic foliation of this class may be topologically equivalent to but a finite number of foliations of the same class, modulo affine equivalence. This property is called \emph{total rigidity}. Recent result of Lins Neto implies that the finite number above does not exceed 240. This is the first of the two closely related papers. It deals with the rigidity properties of quadratic foliations, whilst the second one studies the foliations of higher degree.

Abstract:
The restricted version of the Hilbert 16th problem for quadratic vector fields requires an upper estimate of the number of limit cycles through a vector parameter that characterizes the vector fields considered and the limit cycles to be counted. In this paper we give an upper estimate of the number of limit cycles of quadratic vector fields $"\sigma $--distant from centers and $\ka $-distant from singular quadratic vector fields" provided that the limit cycles are $"\delta $--distant from singular points and infinity".

Abstract:
Let E/Q be an elliptic curve with a fixed modular parametrization F : X_0(N) --> E and let P_1,...,P_r be Heegner points on E attached to the rings of integers of distinct quadratic imaginary field k_1,...,k_r. We prove that if the odd parts of the class numbers of k_1,...,k_r are larger than a constant C=C(E,F) depending only on E and F, then the points P_1,...,P_r are independent in E/(torsion). We also discuss a possible application to the elliptic curve discrete logarithm problem.

Abstract:
In these notes we study hyperplane arrangements having at least one logarithmic derivation of degree two that is not a combination of degree one logarithmic derivations. It is well-known that if a hyperplane arrangement has a linear logarithmic derivation not a constant multiple of the Euler derivation, then the arrangement decomposes as the direct product of smaller arrangements. The next natural step would be to study arrangements with non-trivial quadratic logarithmic derivations. On this regard, we present a computational lemma that leads to a full classification of hyperplane arrangements of rank 3 having such a quadratic logarithmic derivation. These results come as a consequence of looking at the variety of the points dual to the hyperplanes in such special arrangements.

Abstract:
We study the restriction of the Fourier transform to quadratic surfaces in vector spaces over finite fields. In two dimensions, we obtain the sharp result by considering the sums of arbitrary two elements in the subset of quadratic surfaces on two dimensional vector spaces over finite fields. For higher dimensions, we estimate the decay of the Fourier transform of the characteristic functions on quadratic surfaces so that we obtain the Tomas-Stein exponent. Using incidence theorems, we also study the extension theorems in the restricted settings to sizes of sets in quadratic surfaces. Estimates for Gauss and Kloosterman sums and their variants play an important role.

Abstract:
Given a triangulated region in the complex plane, a discrete vector field $Y$ assigns a vector $Y_i\in \mathbb{C}$ to every vertex. We call such a vector field holomorphic if it defines an infinitesimal deformation of the triangulation that preserves length cross ratios. We show that each holomorphic vector field can be constructed based on a discrete harmonic function in the sense of the cotan Laplacian. Moreover, to each holomorphic vector field we associate in a M\"obius invariant fashion a certain holomorphic quadratic differential. Here a quadratic differential is defined as an object that assigns a purely imaginary number to each interior edge. Then we derive a Weierstrass representation formula, which shows how a holomorphic quadratic differential can be used to construct a discrete minimal surface with prescribed Gau{\ss} map and prescribed Hopf differential.