Welcome to XII International ICMAT Summer School on Geometry, Mechanics and Control organised by the Geometry,  Mechanics and Control (GMC) Network.

The twelfth edition of the International Summer School on Geometry, Mechanics and Control will be held at Universidade de Santiago de Compostela, Spain, in July 2-6, 2018 (New location!). The conference venue will be AULA MAGNA in the FACULTAD DE MATEMÁTICAS.

The school is oriented to young researchers, Ph.D. and postdoctoral students in Mathematics, Physics and Engineering, in particular those interested in focusing their research on geometric control and its applications to mechanical and electrical systems, and optimal control. It is intended to present an up-to-date view of some fundamental issues in these topics and bring to the participants attention some open problems, in particular problems related to applications.  This year the courses will be delivered by:

• Eduardo García Río (Universidade de Santiago de Compostela, Spain): GEOMETRY OF THE BACH TENSOR IN DIMENSION FOUR

The Bach tensor of a four-dimensional pseudo-Riemannian manifold is defined by the gradient of the quadratic curvature functional given by the L2-norm of the Weyl tensor. It is a divergence-free, trace-free and conformally invariant (0,2)-tensor field. Since Bach-flat manifolds are the most natural generalization of Einstein metrics, they have received special attention in the literature. Recently Bach-flat metrics have been investigated within the framework of gradient Ricci solitons (another generalization of Einstein metrics motivated by the Ricci flow).

Our first purpose is to introduce the Bach tensor and to point out some of its properties by focusing on homogeneous manifolds. As a consequence one gets a classification of conformally Einstein four-dimensional homogeneous manifolds. Then, turning to Ricci solitons, we show the existence of strictly Bach-flat gradient Ricci solitons in neutral signature (2,2), a situation without Riemannian counterpart. This leads to the consideration of some soliton-like equations in affine surfaces, a topic that will be slightly explored.

• Tom Mestdag (University of Antwerp, Belgium): SYMMETRY AND REDUCTION OF LAGRANGIAN SYSTEMS. SLIDES

The enormous importance of the concept of symmetry in a great number of applications in physics is beyond any doubt. Symmetry properties of mechanical systems in particular have been studied intensively in the last decades. The bulk of the literature, however, concentrates on the Hamiltonian description in which mainly the theory of Poisson and symplectic manifolds plays an important role. Less well-known is the process of reduction by a symmetry group for Lagrangian systems. In the literature, there are in fact different paths that lead to different Lagrangian reduction theories. In this course we will focus on a few of those paths.

We start the course with the relevant machinery from tangent bundle geometry, Lie group actions and connections. We then consider Lagrange-Poincare reduction and its relation to Lie algebroids, and we deal with some related questions such as reconstruction by means of a principal connection, relative equilibria, un-reduction and the inverse problem of Lagrangian mechanics in this context. Next, we consider Noether's theorem and Routh reduction. As an application, we generalize Maupertuis' principle and use symmetry reduction to show its relation to Finsler geometry.

• Catherine Meusburger (FAU Erlangen-Nürnberg, Germany): INTRODUCTION TO POISSON LIE-GROUPS

Poisson-Lie groups can be viewed as the classical analogues of quantum groups and play and important role in integrable systems and in gauge theory. A Poisson-Lie group is a Lie groups that is also a Poisson manifold in such a way that the multiplication is a Poisson map.            

On the  Lie algebra level, this implies that the dual vector space of its Lie algebra also has a Lie algebra structure, and the two Lie algebra structures satisfy a compatibility condition, making it into a Lie bialgebra. Lie bialgebras can therefore be viewed as the infinitesimal counterparts of Poisson-Lie groups. We explain first the relation between Lie bialgebras

We explain first the relation between Lie bialgebras and Poisson-Lie groups, basic constructions  such as duals and doubles and introduce quasitriangular Poisson-Lie groups and Lie bialgebras. As applications, we discuss the relation between Poisson-Lie groups and integrable systems and the role of Poisson-Lie groups in gauge theory. The latter is related to the notion of a Poisson G-space, a Poisson manifold with a Poisson-action of  a Poisson-Lie group G. This leads to a description of the symplectic structure on moduli spaces of flat connections on compact oriented surfaces.

If time permits, we also comment on Poisson reduction for Poisson-Lie groups and  on the role of dynamical r-matrices and Poisson groupoids inthis context.