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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, witha 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 the geometry of differentiable manifolds. There are two active groups in the Institute working in contact topology, symplectic topology and Poisson geometry. Another very active research group is centered around the study of geometric questions in differential equations. There is also research in other special types of geometric structures: metrics with special holonomy, foliations, multi-symplectic structures, path geometries, etc.

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.