Research developed within the Algebra group connects with a variety of (sub)areas of Mathematics and Computer Science. Graph-theoretic, geometric or topological arguments have widespread use.
Automata theory: Descriptional complexity in the average case through the analytic combinatorics of conversion methods between regular expressions and finite automata. Invertibility studies to develop a public-key cryptography system based on linear transducers. Use of pre-grammars for type inhabitation.
Analysis is a key area of mathematics, having enormous applications and relations with other areas of mathematics, physics, engineering, etc. The main strands of the work developed within the analysis group which completely described by the keywords are creation and developments of methods of integral transforms and integral equations with special functions as the kernels, methods of special functions and orthogonal polynomials to solve different analysis problems, numerical analysis and computation.
Computers have fundamentally changed the relationship between mathematics, computing and other sciences. Apart from their invaluable role in numerical, symbolic and experimental applications, computation is per se an important object of mathematical study, constantly proposing new challenges for mathematics.
With the rapid growth of computational power, today's computers provide increasingly better simulation capabilities and require the adaptation of algorithms to new architectures. The importance of computational mathematics, has never been greater.
The theory of Dynamical Systems has its origin in the qualitative study of ordinary differential equations or difference equations. Our research addresses these types of systems, with time varying in a continuous, discrete or complex way. We study algebraic, analytic, geometric, probabilistic and topological properties of systems, with some problems arising in Biology, Economics, and Engineering.
Geometry is a central area of modern mathematics. The 3 main strands of our work (described by the keywords) are closely intertwined, through the use of dynamical (symplectic) methods in the study of geometric problems, on the one hand, and the use of geometric methods in the study of problems in dynamics, on the other. We also contribute with applications to Economics and Biology, in the line Mathematical Models and Applications.
In the following each of the 3 strands of our work is described in more detail.
Many of CMUP's researchers work on applications of Mathematics to other subjects in science and technology. For some, this is the main aspect of their research; for others, applications appear as interesting uses for their mathematical expertise.
The group of Probability and Statistics aggregates researchers from CMUP with the common denominator that they all use tools of probability and statistics to carry out research.
The research subjects covered include ergodic theory, extreme value theory, signal processing and statistical modelling and inference. Moreover, applications to biomedicine, engineering, insurance and meteorology are also studied.
Research in this area ranges from algebraic topics such as semigroups and groups (finite, profinite, or general) to mathematical models used in computer science, namely various flavors of automata and formal languages. Current work aims not only to contribute to the theories of each of the topics but also to explore the connections between them, with applications in both directions.
