Home Research Lines

Research Lines.

The main fields of activity at IUMPA are Algebra, Applied Mathematics, Mathematical Analysis, Mathematical Modelling, and Topology and Geometry.

Algebra

Factorised groups

 
-Research coordinator: Mari Carmen Pedraza Aguilera
The study of factorised groups is a productive area of research in finite group theory, with the structural impact of the factors on the whole group one central question. We address several problems in this context: extensions of the Kegel-Wielandt theorem through p-decomposable groups, or the influence of conjugacy class sizes of elements in the factors. We are particularly interested in factorised groups with certain connection properties between the factors (total, mutual, conditional permutability; L-connection).

Lattice and subgroup embedding properties

 
-Research coordinator: Ana Martínez Pastor
One of the most effective methods in the structural study of finite groups, parallel to the classification of simple groups, consists of the analysis of the interrelationships between relevant families of subgroups and their embedding in the group, as well as their lattice properties. In this frame, we deal with the influence of the normalisers of Sylow subgroups in the structure of the group. Also certain significant subgroup lattices are analysed.

Conjugacy classes and characters in finite groups

 
-Research coordinator: María José Felipe Roman
Character Theory and the study of conjugacy classes in finite groups are two closely related research lines whose main objective is to get information on the structural properties of the group, such as its normal structure, simplicity, solvability,... We are interested in obtaining this information, also in factorized groups. Many results in this frame need knowledge about both working fields in order to be addressed in a satisfactory manner.

Groups: Algebra (Group Theory)

Applied Mathematics

Mathematical physics of periodic and non-periodic structures


-Research coordinator: L. M. Garcia-Raffi
Acoustic Metamaterials: The properties of the periodic structures have been exploited during the last decades to control sound propagation by means of sonic/phononic crystals. However, during the last years several complex materials have extended the concept of order without periodicity. Quasicrystals, fractals materials are examples of such a kind of materials that have introduced new wave physics.  The characterization of these materials has its roots in both optimization methods and some concepts of pure mathematics.
 

Bibliometric Indices in Scientific Research


-Research coordinator: Antonia Ferrer Sapena
The definition of information indices has become a fundamental tool for research assessment, as well as economic indices are central in modern economic theory. The mathematical structure of these indices is related to abstract integration, using universal integrals and integrals in the scalar case, and in general integration with respect to fuzzy capacities. These theoretical questions and its concrete applications for bibliometric analysis of data, journals and in open information, for example, has become the subject of this new line of research.
 

Graph-based databases and Fraud Detection


-Research coordinator: J. M. Calabuig Rodríguez
We use mathematical tools (together with advanced technological software) to analyze and detect fraudulent practices both in public administration and private companies. From a mathematical point of view our results are based in the use of topological (pseudo)metrics defined in graph-based databases. The results are implemented using different technological software as Neo4j and R. As an extension of this scheme we also use the structure of Blockchain.
 

Homogenization and correction of celestial reference frames


-Research coordinator: María José Martínez Usó
The positions and motions on the celestial sphere must be expressed in a well-defined reference system, materialized in different reference frames (multitude of massive catalogs, each with its own characteristics). Their accurate definition, determination, and correction along time is essential to assure a precise positioning in, for example, satellite navigation. The study and homogenization of these catalogs, as well as their adaptation to the future reference frame based on Gaia, is one of the aims of this line of work. We propose methods to improve the integration techniques used in the study of problems linked to orbital motion, both in the case of satellite theory and in the case of the planetary theories of the solar system. In addition, we seek to use old astronomical observations (mainly from Middle Age) to update and improve current astronomical parameters.

  Research Groups: MADPhy: Mathematics and Applications to Data and Physics, Celestial Mechanics and Reference Systems

Mathematical Analysis

Geometry of Banach spaces


  -Research coordinator: Vicente Montesinos Santalucía
Equivalent norms may improve the differentiability and convexity properties. Thus, functions become differentiable at many points, allowing differentiability techniques for solving analytically several problems | shortest or farthest distances to subsets, for example, becomes feasible. The isomorphic characterization of spaces with good renormings is crucial. Lipschitz functions appear naturally in functional-analytical questions. They allow for linearization of non-linear problems. The structure of the Banach space to work with appears then more transparent.

Infinite-dimensional holomorphy


  -Research coordinator: Pablo Sevilla Peris
Holomorphic functions on infinite-dimensional Banach spaces and algebraic structure of such functions and their analytical properties. The power series of such functions on a space with unconditional basis may not converge everywhere. Describing the points of convergence is related with Bohr radius and unconditional basis of the space of homogeneous polynomials. We deal with Banach spaces of Dirichlet series, scalar and vector valued, connecting holomorphic functions and Hardy spaces on the infinite-dimensional torus.

 

Functional Analysis, Topology and applications


  -Research coordinator: Enrique Alfonso Sánchez Pérez
The theory of integration with respect to vector measures has been applied in several areas of functional analysis in recent years: function spaces, approximation, operator theory and Banach lattices. This line of research in Mathematical Analysis explores further applications of this theory, for example in the computation of Optimal Domains of relevant operators. Quasi-pseudo metrics has also been applied for multiobjective optimization, and recently, for using Lipschitz extensions of operators acting in quasi-metric spaces in the context of machine learning.

 

Li-Yorke chaos, distributional chaos, and recurrence properties


-Research coordinator: Alfred Peris Manguillot
We focus on:

1.- Analysis of different concepts of distributional chaos, which do not coincide in non-linear dynamics, but which may coincide in linear dynamics.
2.- The study of scrambled sets for hyperspaces associated with operators.
3.- Other properties, like mean Li-Yorke chaos, ergodic properties, etc, that can be related to distributional chaos.
4.- Recurrence  concerning the study of sets of integers given by the hitting times a fixed orbit intersects arbitrary non-empty open sets.

 

Lineability, algebrability and spaceability


-Research coordinator: J. Alberto Conejero Casares
This line deals with the following problems:

1.- Study of algebrability of the set of hypercyclic functions for the derivative operator.
2.- Algebrability analysis for sets of pathological functions from the point of view of dynamics on the interval.
3.- Search of new techniques for lineability/algebrability/spaceability of sets of functions, sequences, series, operators, etc.

 

Topological and measure-theoretic properties in linear dynamics, and non-linear dynamics


-Research coordinator: Alfred Peris Manguillot
Some precise objectives are:

1.- Study of the shadowing property in linear dynamics, especially for weighted shifts.
2.- Existence of invariant measures for operators with special properties.
3.- Dynamics of polynomials and holomorphic functions of infinite variables.
4.- Sensitivity and transitivity properties via Furstenberg families for general dynamical systems.

 

Semigroup theory and fractional calculus applied to PDE's


-Research coordinator: J. Alberto Conejero Casares
The orbits of dynamical systems assocciated to linear PDE's and to systems of an infinite number of ODE's can be described in terms of semigroups of linear operators, which are like exponentials of the differential operator. With such a representation, chaos can be described in different phenomena like traffic or cell growth of cancer cells. Special attention is devoted to fractional PDE's, which are of increasing interest in the representation of physical and biological process.

 

Operators on functions and sequence spaces


  -Research coordinator: José Bonet Solves
We study weighted Banach spaces of holomorphic or differentiable functions and operators between them. In particular, we study Cesàro and other integral and matrix operators on Banach and Fréchet function or sequence spaces, weighted composition operators on Banach spaces of analytic functions, mean ergodicity and power boundedness of all these operators, spaces of (vector valued) Dirichlet series and solid hull and multipliers of weighted spaces of analytic functions defined by sup norms. Collaborators: A. A. Albanese, W. Lusky, J. Taskinen, W. J. Ricker.

 

Time frequency analysis and applications


  -Research coordinator: Carmen Fernández
We investigate in translation invariant subspaces, Fourier integral operators in modulation spaces and processing of biomedical signals with time frequency analysis. We have been regularly collaborating with the Arrhythmias Unit of the tertiary centre Hospital Universitari Politècnic La Fe (Valencia) about methods to classify the different subtypes of atrial fibrillation, in order to analyze the state of progression of the arrhythmia. Collaborators: P. Boggiatto, J. Toft.

 

Partial differential operators and extension of smooth functions


  -Research coordinator: Enrique Jordá Mora
We investigate on hypoelliptic linear partial differential operators with constant strength in classes of ultradifferentiable functions, pseudo-differential operators, regularity of linear partial differential operators in classes of tempered ultradistributions and the Wigner transform and extension operators for smooth functions from compact subsets. Collaborators: C. Boiti, L. Frerick, A. Oliaro, J. Wengenroth.

 

The separable quotient problem


-Research coordinator: Manuel López Pellicer
The Separable Quotient Problem (open problem). It is well known that small Banach spaces, I mean with denseness less than the bounding cardinal, as well as the big Banach spaces in the sense of Todorcevic, have separable quotient. Our aim is to get new characterizations of the separable quotient problem and to apply these characterizations to Banach Spaces with denseness between the  bounding cardinal and the big cardinal considered by Todorcevic, as well as to apply the new obtained characterizations to proof rapididly the existence of Separable Quotient problem, in known cases with very technical profs.

 

Nikodym properties on rings and algebras


-Research coordinator: Manuel López Pellicer
The aim of this line is two find rings and algebras of subsets for which Nikodym property imply strong Nikodym properties and to apply these results to localization problemas, like ranges of vector measures.

Research Groups: Geometry of Banach spaces. Infinite dimensional holomorphy (GBSIH), MADPhy: Mathematics and Applications to Data and Physics, Dynamics of operators (DYNOP), Functional Analysis, Operator Theory and Time Frequency Analysis, New Functional and Descriptive Methods in Functional Analysis (NUMAF)

Mathematical Modelling

Industrial Mathematics


-Research coordinator: Pedro José Fernández de Córdoba Castellá
The Industrial Mathematics line focuses on the resolution of industrial interest problems. In this area, and as examples, numerical simulation of problems of energy efficiency in buildings, analysis of heating and cooling geothermal systems or studies of heat transfer in LED bulbs have been developed. Moreover, the group is interested on promoting the relationship with companies and especially on transferring mathematical technology to Industry

Mathematical modelling and numerical methods


-Research coordinator: Pedro José Fernández de Córdoba Castellá
This line focuses on developing mathematical models to describe different scientific and social problems and on developing algorithms and computer codes to simulate them. As an example, it focuses on modelling heat and mass transfer problems; on simulating problems of electromagnetic propagation in photonic systems or on optimizing processes in biological systems. These numerical tools include statistical and Monte Carlo techniques, finite differences methods or finite elements methods and modal techniques.

Mathematical Physics


-Research coordinator: José María Isidro San Juan
Current topics of interest at the interface between mathematics and physics constitute the major research line pursued by this group. Specifically, two topical issues are investigated:
    1. Foundations of quantum theory, especially from a thermodynamic viewpoint (the so-called perspective of "emergence of quantum theory");
    2. A differential-geometric approach to thermodynamics, as a byproduct of the above, but also interesting in its own right.
 
   

Biological and biomedical mathematical modeling (B2M2)


-Research coordinator: J. Alberto Conejero Casares
New developments and applications in Biology and Biomedicine are based on the mathematical modeling and simulation of biological systems. Among others, tools from applied mathematics, numerical analysis, network science, statistics, and data science are used for these purposes. Special attention will be devoted to: 1.- Data analysis of information quality and development of subsequent predictive models. 2.- Synthetic biology, where engineering principles are applied to biological circuits (built of proteins and nucleic acids) in cells.
 

Signal processing methods for financial time series


-Research coordinator: Francisco Guijarro Martínez
This line of research concentrates on the study of mathematical tools of signal processing to be applied for the forecast of financial time series (automatic trading of stocks, futures, etc.). In particular: 1.- Detection and optimization of patterns in financial time series. 2.- Development of automatic trading strategies based on patterns and tendency measures.

Research Groups: Interdisciplinary Modelling Group- InterTech, Dynamics of operators (DYNOP)

Topology and Geometry

Asymmetric topology


-Research coordinator: Carmen Alegre Gil
In 1962, Pervin proved that every topological space is quasi-uniformizable. This result increased significantly the interest in asymmetric topological structures (quasi-uniformity, quasi-metrics, asymmetric norms, etc.). The study of these structures has been expanding the last years due not only to its theoretical interest but to their applications in areas like Computer Science. The overall objective of our reseach is to study properties of the asymmetric topological structures that may contribute to development of these applications.

Fixed point theory


-Research coordinator: Pedro Tirado Peláez
The aim of our research is to obtain fixed-point theorems in the context of quasi-metric spaces as well as to study the problem of characterizing several different notions of quasi-metric completeness, which appear due to the lack of symmetry, in terms of the existence of fixed-points. Moreover, we also study these problems in the framework of fuzzy quasi-metric spaces.
 

Fuzzy topological structures


-Research coordinator: Jesús Rodríguez López
Fuzzy theory arose as a mathematical attempt to model uncertainty situations and this lead to the birth of the many-valued logic. In this way, we are interested in studying some connectives of this logic like t-norms, t-conorms, fuzzy implications, etc. Furthermore other mathematical areas like topology has been influenced by the fuzzy theory. We also deal with fuzzy mathematical structures generating (fuzzy) topologies as fuzzy metrics, fuzzy uniformities or fuzzy proximities.

Fuzzy and metric tools for image modelling


-Research coordinator: Samuel Morillas Gómez
Ranging from low-level image processing tasks such as image smoothing or filtering to high-level pattern recognition and image understanding problems, fuzzy logic is employed to apprach different challenges in the field. Lately, the development of abstract image models able to provide solutions to practical applications as well as visual image understanding is being of special interest. We approach this problem from a fuzzy point of view using different fuzzy tools, including fuzzy metrics, for image modelling.

Algebraic Geometry, Singularity Theory and Applications


-Research coordinator: Francisco José Monserrat Delpalillo
We work on several research areas within Singularity Theory and Valuation Theory using, as main tools, methods of Algebraic Geometry. This includes topics as convex cones associated with algebraic varieties, multiplier ideals, plane valuations and Newton-Okounkov bodies. In addition, we consider applications to related areas as Theory of Error-Correcting Codes and Theory of Algebraic Foliations.

Complex Analytic Singularities


-Research coordinator: Carles Bivià Ausina
We study numerical invariants attached to complex analytic maps and sets. These invariants are based on instersection theory and therefore leads to the study of fundamental notions like the integral closure of ideals and submodules of a free module. The description of numerical invariants and integral closures in terms of Newton polyhedra is one of our main subjects of interest. We also study the geometry of complex polynomial maps and the Jacobian conjecture.

Research Groups: Topology and its Applications, Singularity Theory and its Applications
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