The parameterization method for invariant curves associated to parabolic points

Treballs finals del Màster en Matemàtica Avançada, Facultat de matemàtiques, Universitat de Barcelona, Any: 2018, Director: Ernest Fontich Julià[en] In the first part of this work we present the parameterization method for invariant manifolds and we apply it to prove the existence of stable invariant curves of planar maps associated to a fixed point with an eigenvalue $\lambda$ such that $0 < |\lambda| < 1$. We study both the case in which the map is analytic and the case in which it is differentiable. In the second part we apply the parameterization method to obtain the existence of a stable analytic curve associated to a nilpotent parabolic fixed point of an analytic map. The main result of this master thesis is the existence of such a stable curve. Finally, we perform a numerical simulation in order to estimate the growth of the coefficients of a parameterization of this curve

Almost periodic solutions for an asymmetric oscillation

In this paper we study the dynamical behaviour of the differential equation \begin{equation*} x''+ax^+ -bx^-=f(t), \end{equation*} where $x^+=\max\{x,0\}$,\ $x^-=\max\{-x,0\}$, $a$ and $b$ are two different positive constants, $f(t)$ is a real analytic almost periodic function. For this purpose, firstly, we have to establish some variants of the invariant curve theorem of planar almost periodic mappings, which was proved recently by the authors (see \cite{Huang}).\ Then we will discuss the existence of almost periodic solutions and the boundedness of all solutions for the above asymmetric oscillation.Comment: arXiv admin note: substantial text overlap with arXiv:1606.0893

The Loewner equation: maps and shapes

In the last few years, new insights have permitted unexpected progress in the study of fractal shapes in two dimensions. A new approach, called Schramm-Loewner evolution, or SLE, has arisen through analytic function theory and probability theory, and given a new way of calculating fractal shapes in critical phenomena, the theory of random walks, and of percolation. We present a non-technical discussion of this development aimed to attract the attention of condensed matter community to this fascinating subject

Hilbert schemes of points on a locally planar curve and the Severi strata of its versal deformation

Let C be a locally planar curve. Its versal deformation admits a stratification by the genera of the fibres. The strata are singular; we show that their multiplicities at the central point are determined by the Euler numbers of the Hilbert schemes of points on C. These Euler numbers have made two prior appearances. First, in certain simple cases, they control the contribution of C to the Pandharipande-Thomas curve counting invariants of three-folds. In this context, our result identifies the strata multiplicities as the local contributions to the Gopakumar-Vafa BPS invariants. Second, when C is smooth away from a unique singular point, a special case of a conjecture of Oblomkov and Shende identifies the Euler numbers of the Hilbert schemes with the "U(infinity)" invariant of the link of the singularity. We make contact with combinatorial ideas of Jaeger, and suggest an approach to the conjecture.Comment: 16 page

A support theorem for nested Hilbert schemes of planar curves

Consider a family of integral complex locally planar curves. We show that under some assumptions on the basis, the relative nested Hilbert scheme is smooth. In this case, the decomposition theorem of Beilinson, Bernstein and Deligne asserts that the pushforward of the constant sheaf on the relative nested Hilbert scheme splits as a direct sum of shifted semisimple perverse sheaves. We will show that no summand is supported in positive codimension