Lines of research

Commutative Algebra and Algebraic Geometry

In Algebraic Geometry the main goal is to study, from several points of view,  the geometry of the varieties defined by polynomical equations and some attached algebraic objects, including the study of vector bundles defined over these varieties, and also the moduli spaces that classify them. Moreover complexes of sheaves and stability conditions in triangulated categories are studied. From a more arithmetic approach, there is activity on Arakelov theory. Some of the considered varieties in our group, have some specific properties. For instance, abelian varieties which are varieties with a group structure; irregular varieties which are close to abelian varieties; and toric varieties which have the action of a group. Another main subject are Hilbert schemes which classify subvarieties of some given variety. From a computational point of view  there is activity on Elimination theory and effective methods applied to Algebraic Geometry.

In Conmutative Algebra several fundamental and applied aspects are considered, mainly in relation with Combinatoric Algebra. More precisley there is activity in homological methods in positive characterictic, the algebraic Mellin transform, and the estructure of Gorenstein rings in any dimension. Also in the modern study of syzygies and invariants attached to free resolutions. In a more applied context, several problems related to semigroups and the polynomials attached to orders, graphs and matroids are considered. 

 

Stochastic Analysis

Research pivots in a balanced way between fundamental research and the transfer of knowledge to the financial sector, articulating in two strands: the theory of stochastic partial differential equations (sPDE) and the study of continuous time models in financial markets.

Topics covered in sPDE are varied, such as probabilistic potential theory, stochastic wave equations with nonlinear coefficients, sPDE with fractional noise, and Malliavin calculus.

When it comes to studying financial markets, stochastic analysis tools are applied to address issues such as the equilibrium problem when there are investors with asymmetric information, financial bubble models, hybrid product valuation, and volatility models. . fractionated. Financial risk departments could be the professional destination of doctoral students trained in this area.

 

 

Mathematical Analysis

This research line includes Mathematical Analysis and Partial Differential equations (PDE).

The main subjects that are considered in Mathematical Analysis are  classical problems in potential and in operator theory, mainly in several complex variables, of potential theory, as well as geometric measure theory and of harmonic analysis, including the theories of quasi-conformal mappings and of singular integrals. Some of these tools are used to study random point processes that arise in some models of fermions and in the spectral description of random matrices.

The research interests in Partial Differential Equations are also focused on quite diverse problems, including  fluid mechanics, elliptic equations, geometric and functional inequalities, or spectral problems by combining techniques from calculus of variations, harmonic analysis, geometric measure theory, numerical computations, and computer-assisted proofs. 

 

Computer Science

This line of research studies theoretical aspects related to the proposals of new Artificial Intelligence models and techniques as well as practical aspects that study their application to problems in different fields such as medicine, finance or linguistics. Specifically, the Faculty members involved in the Master work in areas such as machine learning, deep learning, computer vision and multi-agent systems, as well as computer graphics, interactive systems, scientific and data visualisation.

The research line is supported by a wide experience in European projects, as well as in the National R&D Plan, Challenges and Excellence projects, as well as in scientific production in high impact journals and international conferences of recognised prestige. In addition, the teaching staff participates in different university and interuniversity master’s degrees (Artificial Intelligence, Biomedical Engineering, Data Science, Biomedical Data Science, Computer Vision) specific to the research area.  

Differential Geometry and Topology

 

 This research direction includes several topics in Geometry, such as gauge theories, Higgs bundles, moduli spaces of geometric structures, and group actions on manifolds. It also encompasses a number of topics of Algebraic Topology: higher homotopical structures, rational and p-adic homotopy, derived categories, and topology of complex algebraic varieties. Hodge theory relates important problems in Geometry and Topology, such as the existence of almost-complex structures or the calculation of their invariants. In another vein, topological data analysis is an innovative and powerful technique for applications of Topology to Neural Networks and Data Science, which in addition it is a new way of transferring knowledge to companies and research centers working with large volumes of data.

 

Mathematical Logic

The main research interests are in Algebraic Logic (logical systems like fuzzy, modal and intuitionistic logics, and metalogical problems with tools of universal algebra and category theory), Model Theory (definability issues in classical Mathematics, mainly in the setting of Stability Theory and its generalizations), Proof Theory (proof systems and the computational and constructive content of proofs), Set Theory (large cardinals, combinatorics and forcing), foundations and in set-theoretic topology (cardinal sequences for Boolean algebras and for Lindelöf P-spaces).

 

Dynamical Systems

theoretical as well as computational, including applications to other sciences. The topics covered are Hamiltonian systems and non-conservative system, both in low and infinite dimension, continuous  dynamical systems (described by differential equations) and discrete  dynamical systems (generated by iteration of functions). Special emphasis is placed on celestial mechanics and astrodynamics, and the design of space missions.

In Holomorphic Dynamics, aspects of the theory of iteration of holomorphic (or meromorphic) functions in the complex plane are worked on. Topics of special interest are the dynamics of transcendental mappings, root-finding algorithms as dynamical systems, quasiconformal surgery as a tool in complex dynamics, and singular perturbations of rational maps.

 

Number Theory

The main research area is Arithmetic Geometry, studying the relation between geometric objects defined over number fields such as elliptic curves, abelian varieties and algebraic varieties and modular and automorphic forms. This is part of a network of conjectures known as the Langlands Program, which includes reciprocity (or modularity) conjectures and Langlads functoriality. Another subject covers Galois representations, with applications to the Inverse Galois Problem. Other than the Langlands Program, there is interest on the Sato-Tate conjecture, Differential Galois Theory, diophantine equations and cryptography.

In boldface, the corresponding members in the academic commission