RESEARCH LINES
1. Abstract evolution equations
The main goal of this part is to address new open problems, that are of current interest for researchers in the area of Analysis of Abstract Evolution Equations, that is the analysis of processes which develop in discrete or continuous time. Our goal is to understand the qualitative behaviour of solutions of linear and nonlinear evolution equations which are important in areas such as physics, applied mathematics, and many others.
We will focus on two general topics: Theory and Applications. In the line of theory, we will rely on important properties of strongly continuous families of bounded and linear operators. We will closely examine specific classes of these families,which are crucial in current developments of both theoretical issues and practical applications of Abstract Evolution Equations. Concerning applications, we will concentrate on the development of two sub topics of research: The first, is the study of qualitative properties of partial integro-differential equations that can be modeled as an evolution equation, linear or nonlinear, in the context of Banach spaces. The class of abstract evolution equations that we have in mind contain important well-known models such as: The Westervelt, Kusnetzov and Jordan-Moore-Gibson-Thomson equations, among others. Also, we will be interested in nonlocal versions of them.
2. Non linear analysis in infinite dimensions: Dirichlet series and algebras of holomorphic functions
The main point of this line of work is to produce advances in the classical settings of harmonic analysis and complex analysis of one and several variables, by using an infinite dimensional point of view. Many lasting open problems about Dircihlet series and the Bohr radius have been recently solved and their solution are relevant and also related with fields like analytic number theory, functional analysis, Banach spaces, probability theory, Fourier analysis, and complex analysis of several and infinite variables. For instance, recent advances about mutidimensional Bohr radius can be used to determine domains of convergence for several holomorphic function spaces and vector Dirichlet series.
3. Dynamics of operators and applications
Concerning dynamics of operators, we will focus our attention on the study of several problems that arise on operator theory as hypercyclicity, ergodicity, recurrence, mixing and chaos. We want to study the relationship between combinatorial number theory and the dynamics of operators. We want also to study triangular maps that combine a non-linear part with a linear one, and non-autonomous dynamical systems. Another goal we have is to combine our knowledge on the dynamics of operators with techniques of fuzzy and hyperspace dynamics to offer a new line of research in linear dynamics.
Finally, we will look for applications of dynamical systems techniques to mathematical models of epidemiological diseases, and other applications.
4. Time-frequency analysis and applications in signal processing
Time-frequency analysis consists of using translations and modulations to analyze functions and operators. It is a type of Fourier analysis that simultaneously and symmetrically treats time and frequency. Our first aim in this line is to use of techniques from time-frequency analysis to study Gelfand-Shilov spaces, which can be considered as analogous to the Schwartz class in the context of classes of ultradifferentiable functions. We also plan to study the dynamics and spectrum of composition operators in the Gelfand-Shilov classes.
Our second aim in this line is to apply time-frequency analysis to signal processing. The study of biomedical signals is one of the areas of major interest. In particular, analysis of recordings acquired by non-invasive methods can be of great use to provide an aid to clinical diagnosis. Other applications in signal processing of time-frequency analysis in which we are interested are fault diagnosis in electrical machines and pattern detection in financial time series.
