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LINE 1
Algebraic Geometry and Mathematical Physics, Moduli Spaces and Group Theory

In this line we cover topics of Algebraic Geometry, and of its tool which is Algebra, and its applications which nowadays go into various fields of Physics. As subjects in pure Algebraic Geometry, some of us propose in the sub-line described below a systematic study of singularities centered in the topology of the Milnor fibration of them. As subject in pure Algebra, there is a study of group theory in the sense we describe below. A little more suprising may be the work in Algebraic Geometry as related to physics, since, although Physics had from its beginning an evident relationship to Mathematics, the close links to Algebraic Geometry is a feature of only the last few decades: Physical potentials correspond to connections, and the force fields derived from them correspond to the curvature of the connections, so that force fields satisfying the equation imposed in Physics correspond to the points of algebraic varieties called moduli spaces of stable bundles. This point of view is the main inspiration of the subline ''Moduli Spaces and Mathematical Physics'', in which several of the proposed subjects involve the study of such moduli spaces, and other are related to an emergent application of number theory.

Singularity Theory

Our research involves the study of algebraic and analytic singularities from different perspectives. Specific topics are the vanishing cohomology of singularities with its D-module and Hodge structure, the topology of the Milnor fibrarion, the embedded topology of the link, equisingularity, resolution in characterisric zero and p (including algorithmic aspects, hypersesolutions and descent categories, versal deformations and their base spaces with additional structure, invariants of surface singularities and of complex analytic spaces, rational cuspidal curves, arrangements, arcs spaces and motivic integration).

Moduli Spaces and Group Theory

In the last three decades, there has been many advances in Algebraic Geometry (and other areas of Geometry and Topology) inspired by ideas coming from High Energy Physics. For instance, Donaldson theory. Thanks to the Hitchin-Kobayashi correspondence, the objects of these theories can be interpreted as stable vector bundles. This has inspired the study of the moduli space of vector bundles and related objects (Higgs bundles, holomorphic pairs coming from the vortex equation, etc...). We have been very much involved in the construction of some of these moduli spaces, the study of its geometric and topological properties, and applications to the study of representations of the fundamental group. This area has very strong ties with line I (Differential Geometry, Symplectic Geometry and Geometric Mechanics), for instance, through Gromov-Witten theory and Mirror symmetry. More recently, there have been advances in the Geometric Langlands Program, which relates Algebraic Geometry with ideas coming from Physics and Number Theory.

LINE 2
Differential Geometry, Symplectic Geometry and Geometric Mechanics

The research of the group is divided into two lines, one corresponding to the differential geometric aspects of symplectic geometry, and the other centering on its applied aspects, with a focus on geometric mechanics and control theory. The first line fits broadly into the theme of the study of the global aspects of manifolds, and include topological properties of symplectic and contact manifolds, manifolds with special holonomy, rational homotopy theory of differentiable manifolds, and geometric structures of non-Riemannian type (path geometries, ...). The second line of research focuses on geometric mechanics and control theory. Research themes include geometric field theories, Poisson geometry (groupoids, algebroids, ...), symplectic integration and numerical linear algebra (algorithms for matrix computations), and non-linear dynamics (matrix analysis, matric polynomials, ...). A common interest in gauge theoretic problems and techniques unites the two lines of research.

Symplectic and differential geometry

The research in this sub-line deals with global aspects of manifolds. It includes topological properties of symplectic and contact manifolds, manifolds with metrics with special holonomy, rational homotopy theory of differentiable manifolds, and geometric structures of non-Riemannian type (e.g. path geometries).

Geometric Mechanics and Control Theory

Geometric mechanics involves the study of Lagrangian, Hamiltonian mechanics and control theory using geometric and symmetry techniques. The main guiding idea in its development consists in applying the techniques and methods of differential geometry to the study and description of this kind of systems (classical, field theoretical, or quantum). It can be applied to design new structured algorithms for the numerical study of these systems that capture their essential qualitative behaviours and improve the stability properties of the existing methods.

LINE 3
Mathematical Analysis, Differential Equations and Applications

The institute has several researchers working in this line in a wide spectrum of subjects that covers some theoretical subjects like harmonic analysis or some basic aspects of the theory of partial differential equations until some subjects that are close to physical applications, in particular to some problems of fluid mechanics. Some of the researchers of this research line are working in the numerical simulation of partial differential equations. One of the main strengths of this research line is the group of researchers working in Harmonic Analysis and Partial Differential Equations. There exists an active group, that has a wide international recognition, whose work lies in the interface between these two fields. There are also strong researchers working in Fluid Mechanic problems and some of the newly incorporated researchers have already worked and are interested in the development of an active research line of the mathematical theory of kinetic equations as well as related problems of statistical physics.

Mathematical Analysis and Partial Differential Equations

Mathematical analysis is a corner stone in the solution of many problems coming from the natural and social sciences. During the last century, it has been deeply influenced by the long standing problem of convergence of Fourier series, giving rise to new methods in functional analysis, harmonic analysis and probability. One of the main motivations for mathematical analysis is the study of partial differential equations which model physical phenomena, and this will continue to play a crucial role in the foreseeable future.

Differential Equations and Applications

Differential equations have been used since they were introduced in Mathematics to describe different question of the natural sciences. This research line includes several researchers who are motivated in part or all their work in the study of ordinary and partial differential equations whose study is justified for their usefulness to study some physical or biological question.. In a more precise manner, the researchers in the research subline are working in the mathematical theory of the equations of fluid mechanics, in the analysis of singularities in partial differential equations, in the numerical computation of the solutions of fluid mechanics equations, as well as in mathematical modelling of problems arising in the physics of electrofluids. Some of the research developed in the last years includes also the study of the mathematical theory of kinetic models and had began to consider problems related to mathematical biology.