♦ Moduli spaces and mathematical physics
o PI: Óscar García Prada
This group is devoted to the study of moduli spaces and their relation with various geometric structures. This study involves algebraic geometry, differential geometry, topology, algebra and theoretical physics. We have close connections with other international research networks:
· GEAR: Representations of surface groups, Teichmüller theory and related areas. Funded by NSF (4 years).
· ITGP: Low-dimensional topology and geometry, gauge theory, quantum topology, symplectic topology and geometry, string theory, etc., funded by ESF (2009-2014).
· French-Spanish Network on Geometric Analysis. Funded by CSIC, CNRS and GDRI (international research group), includes several universities from France and Spain.
♦ Singularity theory
o PI: Javier Fernández de Bobadilla
Singularity theory is a transversal research subject in which many different techniques (algebraic, analytic, geometric, topological) converge, from different perspectives (local, global), but with a very close interconnection.
♦ Symplectic geometry, Geometric Theory of PDE and dynamical systems and dynamics
o PI: Francisco Presas Mata
This line is working on outstanding problems related with symplectic geometry and topology and closely related problems (contact, Poisson, multisymplectic geometry). It is focused on a study of fundamental importance for the physical implications of symplectic geometry: dynamics, and the relation between the classical and the quantum level. It is also focused on the study of geometric and topological properties of solutions to differential equations, both ordinary (dynamical systems) and partial.
♦ Geometric Mechanics and control theory
o PI: David Martín de Diego
The main objective of the research is focused on the domain of geometric mechanics and control. The geometrical structures shared by mechanical and control systems include symplectic, contact, Poisson and Jacobi structures as well as other algebraic-geometric structures like Lie algebroids and groupoids, etc., but they still require further study, such as those aspects related to symmetry and reduction and infinite dimensional systems like media with microstructure. At the same time, the study of geometrical integrators suggests that there is a natural geometrical framework for these integrators (Lie groupoids).
♦ Harmonic Analysis
o PI: Javier Parcet
Classical problems. Several members of our group are working actively on the solution of questions around Kakeya's conjecture, Bochner-Riesz multipliers, directional maximal operators, the Kato conjecture and elliptic PDE's, two-weight inequalities for the Hilbert transform, restriction of the Fourier transform.
Noncommutative extensions. The goal is to develop a noncommutative form of Calderón-Zygmund theory to face widely open fields, like convergence of Fourier series and norm estimates for Fourier multipliers in the compact dual of arbitrary discrete groups.
Deep interdisciplinary connections. This is perhaps the most promising aspect of our work. Beyond applications to nonlinear PDE's, number theory or quantum mechanics.
♦ Partial Differential equations
o PI: Diego Córdoba
We will develop further the interplay between ideas from geometric function theory and the calculus of variations on one hand with the theory of weak solutions of non-linear partial differential equations and on the other, with several versions of Calderón problem. We will also incorporate state-of-the-art techniques in harmonic analysis to tackle well known problems in scattering theory, on the development of recovery algorithms and existence, uniqueness, regularity and formation of singularities in equations that arises in fluid mechanics. Our aim is for this research eventually to yield results in concrete applications in Impedance, Diffraction, thermoacoustic tomography and to give rigorous proofs of the existence of singularities.
♦ Dynamical systems and numerical analysis
o PI: Aníbal Rodríguez Bernal
The Lagrangian Approach to Fluid Transport: Dynamical systems theory has in recent years provided a strong framework for describing transport in time dependent flows. The goal is to obtain new developments on the computation of Lyapunov vectors and exponents for non-uniformly hyperbolic trajectories in 2D time dependent flows.
Instabilities in Fluid Mechanics: Bifurcation theory has proven to be a useful tool for the analysis of instabilities in geophysical fluid flows. Mantle dynamics and plate tectonics are studied from the perspective of fluid convection.
Infinite dimensional dynamical systems: The theory of non-autonomous dynamical systems has opened up new possibilities in the analysis of classical models.
Applications to Mathematical Biology: Numerical methods for free boundary problems in fluid mechanics and problems in Mathematical Biology. These include Boundary element methods, finite element methods, Montecarlo Methods and spectral methods.
♦ Number Theory (Analytic and Combinatorial Number Theory, Algebraic Number Theory)
o PI: Javier Cilleruelo
Analytic and Combinatorial Number Theory. One of the features of the analytic and combinatorial number theory is the interplay of a great variety of mathematical techniques, including combinatorics, harmonic analysis, probability theory, algebraic geometry or ergodic theory. The modern analytic number theory has benefitted from the harmonic analysis in some groups related to automorphic forms, while Additive Combinatorics is a relatively recent term coined to comprehend the developments of the more classical combinatorial number theory, mainly focussed on problems related to the addition of integers.
Algebraic Number Theory. Many problems in arithmetic can be translated into the problem of deciding if an algebraic variety contains rational points. Modular varieties and modular curves appear in the proof of three cornerstones in this area: Faltings proof of Mordell's conjecture; Wiles proof of Fermat's last theorem using the Shimura-Taniyama-Weil conjecture, and the Birch and Swinerton-Dyer conjecture whose case of analytic rank one has been proved by Gross-Zagier and Kolyvagin. The concept of height also plays a central role; it appears in two of these proofs and has been used to design algorithms to solve diophantine equations. The Arakelov theory provides a framework to give precise definitions of heights and to study its properties.
♦ Group theory
o PI: Andrei Jaikin
The research in this line deals with the global aspect of finite and infinite groups. Our main theme is the Asymptotic Group Theory. The group maintains close collaboration with researchers belonging to other centers, both within Spain and abroad (in particular the United Kingdom, Germany, US, Israel, and Italy among others).