| Summary: |
The research conducted in CMUP covers a large spectrum of pure and applied Mathematics. To help us to provide a concise view of the breadth of these activities, they will be organized in four main groups, namely: Algebra, Analysis, Geometry and Probability and Statistics. We emphasize, however, that interactions among CMUP members happens at various levels and there is a significant volume of research lying in the borderlines between the traditional interests of the mentioned groups.
The Algebra group will pursue research in semigroup theory including the elaboration of profinite tools for finite semigroups in order to understand the structure of relatively free profinite semigroups over large pseudovarieties and to establish tameness-like properties. Special classes of monoids will be studied from the algorithmic and the structural point of view. Applications of automata theory to geometric group and combinatorial semigroup theory and the development of algebraic tools for applications to combinatorics will be carried out as well as designs of cryptographic systems based on finite and cellular automata shall be invented. Furthermore we aim to find decidability procedures for Kleene algebras and to study the complexity of formal languages. Hyperplane arrangements, parking functions and Young tableaux will be studied as well as some combinatorial decision/optimization problems. Algebraic structures for the automatic generation of computer aided student assessment is intended to be found. Invariant subspaces under graph adjacency matrices action will be described theoretically and arithmetically with applications to coupled cell network. The homology and representation theory of quantum algebras shall be studied as well as generalized quotients of Hopf algebras and depth/normality of ring extensions.
The Analysis group will pursue research in the following themes. Concerning the theory of integral transforms and special functions, polynomial problems of the C  |
Summary
The research conducted in CMUP covers a large spectrum of pure and applied Mathematics. To help us to provide a concise view of the breadth of these activities, they will be organized in four main groups, namely: Algebra, Analysis, Geometry and Probability and Statistics. We emphasize, however, that interactions among CMUP members happens at various levels and there is a significant volume of research lying in the borderlines between the traditional interests of the mentioned groups.
The Algebra group will pursue research in semigroup theory including the elaboration of profinite tools for finite semigroups in order to understand the structure of relatively free profinite semigroups over large pseudovarieties and to establish tameness-like properties. Special classes of monoids will be studied from the algorithmic and the structural point of view. Applications of automata theory to geometric group and combinatorial semigroup theory and the development of algebraic tools for applications to combinatorics will be carried out as well as designs of cryptographic systems based on finite and cellular automata shall be invented. Furthermore we aim to find decidability procedures for Kleene algebras and to study the complexity of formal languages. Hyperplane arrangements, parking functions and Young tableaux will be studied as well as some combinatorial decision/optimization problems. Algebraic structures for the automatic generation of computer aided student assessment is intended to be found. Invariant subspaces under graph adjacency matrices action will be described theoretically and arithmetically with applications to coupled cell network. The homology and representation theory of quantum algebras shall be studied as well as generalized quotients of Hopf algebras and depth/normality of ring extensions.
The Analysis group will pursue research in the following themes. Concerning the theory of integral transforms and special functions, polynomial problems of the Casas-Alvero type and the Hilbert and Hartley transforms on the positive half-axis will be studied. We aim to construct the theory of integral transformations with arithmetic functions in the kernels and to study new equivalences to the Riemann Hypothesis. The large scale transport coefficients, derived by multi-scale techniques, in two-dimensional incompressible flows will be studied and we intend to write a Pontryagin maximum principle applied to some problems. Novel projection-type methods for nonlinear equations will be developed. Furthermore, a numerical library will be developed to compute approximate solutions and the corresponding error bounds, for certain type of nonlinear integral equations, will be developed. Supporting the Computational Mathematics research line, the potential of symbolic computations performed by Mathematica software will be explored to clarify mathematical properties in the domain of orthogonal polynomials in contexts where the complexity of the calculations so requires. Using novel fractal dimension estimator algorithms for finite samples of 2D curves against well-established algorithms, such as rescaled-range-analysis or detrended-fluctuation-analysis, fractal characteristics in real-world financial signals will be studied.
The Geometry group will conduct research in the following directions: geometry and dynamics of real and complex differential equations, algebraic geometry and symplectic geometry. In real dynamics, there are the problems related to 'bifurcations', e.g. heteroclinic networks, where complicated dynamical phenomena may occur. In th |