Research Lines.

The main fields of activity at IUMPA are Algebra, Dynamical Systems, Fuzzy Mathematics, Geometry, Mathematical Analysis, Mathematical Methods in Synthetic Biology, Mathematical Physics, Numerical Methods, Operations Research, Photonics and Topology

Algebra is an important branch of mathematics concerning with abstract structures that emerge as a generalization of ordinary arithmetic operations and their invariants. Although abstract algebra has being traditionally considered an area of pure mathematics, it presents various applications in other branches of science and technology such as physics, chemistry, computer science or more specifically cryptography and code data transmission.

IUMPA has a group of researchers working on Algebra who are dealing mainly with group theory and its applicactions, supported by a competitive project from the Spanish Ministry (MICINN), coordinated with other groups of the Universitat de València and the Universidad de Zaragoza. Some of these researchers also collaborate and are involved with other external research projects.

Essentially, the main interest of the algebra team focuses on the internal structure of finite or infinite groups (with finiteness conditions) through their actions, their representations, their factorizations, the lattices of their subgroups and arithmetical properties. The interest from the applied side is the interaction of groups and semigroups with current problems in the context of automata and formal languages.

**Groups:**Algebra Research Group (Group Theory)

We are mainly concerned about the study of the singularities of maps of several complex variables defined over complex analytic varieties, therefore our research topics belong to singularity theory and intersect fields like algebraic geometry, complex analytic geometry and commutative algebra.

We study several kinds of algebraic invariants attached to maps defined over varieties, we are interested in the relation between these invariants and the effective computation of them. We also analyze the meaning of the constancy of such invariants in deformations of maps in terms of several equivalence relations of maps (for instance, topological trivialy, Whitney equisingularity and bi-Lipschitz equivalence).

^{th}century and is connected with problems in physics. Although mathematical analysis is an area of theoretical research, it has recently undergone a profound transformation, and nowadays it is a useful tool for digital image process and resizing or scaling process. Large and well-recognized leading experts work at IUMPA on different areas of mathematical analysis. They are in charge of six research projects of the Ministry of Education and Science (MEC). The research interests of the group span topics such as real and complex analysis, harmonic analysis and especially functional analysis and operator theory. In the more theoretical aspect many abstract spaces are studied, for instance: locally convex, vector measure, spaces of continuous, differentiable or analytic functions and Fréchet spaces. Also, the geometry of Banach spaces, partial linear differential equations, hypercyclicity, linear chaos of operators and semigroups, descriptive set topology and infinite-dimensional holomorphy are studied.

**Research Groups:**Interdisciplinary Modelling Group- InterTech

**Research Groups:**Interdisciplinary Modelling Group- InterTech

**Research Groups:**Vehicle Routing Optimization and General Systems Theory

Problems of nonlinear wave propagation of light in micro and nanostructured spaces (on either temporary or spatial domain) have a great technological interest. The replacement of the present electronic technology with a new technology based on photonic circuits is one of the great challenges of nanotechnology. This area develops both mathematical models and innovative numerical algorithms, in order to deal efficiently

**and creatively with the analysis of these systems.**

The field of cold atoms and matter waves includes the study of the behaviour of matter under temperatures near absolute zero. This framework upraises new quantum phenomena, analysed by nonlinear equations (similar to optics nonlinear equations). The present experimental control of these systems at low temperatures allows foreseeing new technological applications arising from quantum nature of matter at the atomic scale. Concepts such as nanotechnology or quantum computing are involved in these applications. In this way, our experience in nonlinear optical modelling is used to propose new ideas to this promising field.

**Research Groups:**Interdisciplinary Modelling Group- InterTech

**Research Groups:**Interdisciplinary Modelling Group- InterTech

Its origin lies in Cantor papers published between 1879 and 1884 concerning with problems of uniqueness for trigonometric series. At the beginning of the 20th century, Fréchet and Riesz introduced independently, the concept of abstract space endowed with a topological structure. However the first satisfactory definition about topological spaces using the concept of neighbourhoods was given by Hausdorfff in 1914.

Until the middle of the 20th century, metric compact spaces and uniform spaces attracted the attention of the researchers in this area. In 1962, Pervin proved that every topological space is quasiuniformable. This important result increased significantly the interest in asymmetric topological structures (quasiuniformities, quasimetrics, quasinorms, etc.). In recent years, these structures have had interesting applications in algorithmic complexity, computational programming, computational biology, etc.

IUMPA has a group of researchers dedicated to topology and its applications with two existing competitive projects (MEC, coordinated with a group of the “Universidad del País Vasco” and other group funded by “Generalitat Valenciana”). The main research topics are: asymmetric structures, the hyperspaces, metrics and fuzzy quasimetrics, as well as connections to algebraic structures and applications to various fields of the computer science (complexity of algorithms and programs, semantics, databases, etc.).

**Research Groups:**Topology and its Applications

Wave Propagation in Periodic Media and Acoustics

**Research Group:**FUNAPHY