**Program**

## Mini course Speakers

**Miguel Ángel Javaloyes**(Universidad de Murcia, Spain)

**Finsler Geometry: Riemannian foundations and relativistic applications**, Abstract & References; Slides 1; Slides 2; Slides 3

**Bronislaw Jakubczyk**(Institute of Mathematics, Polish Academy of Sciences, Poland)

**Vector fields, control, and input-output systems**, Abstract & References

## Plenary Speakers

**Alice Barbara Tumpach**(Lille University, France)

**Banach Poisson Lie-group**

We develop a notion of generalized Banach Poisson--Lie group suitable to treat a particular example related to the KdV hierarchy. We construct this Banach Poisson--Lie group structure on the unitary restricted Banach Lie group acting transitively on the restricted Grassmannian. A ``dual'' Banach Lie group consisting of (a class of) upper triangular bounded operators admits also a Poisson--Lie group structure of the same kind. We show that the restricted Grassmannian inherits a Bruhat-Poisson structure from the unitary Banach Lie group, and that the action of the dual Banach Lie group on it (by ``dressing transformations'') is a Poisson map. This action generates the KdV hierarchy, and its orbits are the Schubert cells of the restricted Grassmannian. However, at the level of the groups, the dressing actions do not make really sense because of the lack of Iwasawa decomposition.

**Julio I. de Vicente**(Universidad Carlos III de Madrid, Spain)

**A non-trivial order in the set of multipartite entangled quantum states**

Quantum resource theories provide a rigorous framework to order and quantify quantum states as resources in the context of quantum information theory. These theories are built by specifying a subset of the class of completely positive maps, which can be understood as the dynamics that can be freely implemented under the physical restrictions that arise in the manipulation of the given resource. This set of free transformations induces then naturally an order relation in the set of quantum states. Entangled states are known to play a prominent role in quantum information tasks and entanglement theory is usually formulated as a quantum resource theory in which the free operations correspond to the class of completely positive maps known as local operations and classical communication (LOCC). This defines a partial order among bipartite states that makes it possible to identify a maximally entangled state, which turns out to be the most relevant state in applications. However, the situation changes drastically in the multipartite regime. Not only do there exist inequivalent forms of entanglement forbidding the existence of a unique maximally entangled state, but recent results have shown that LOCC induces a trivial ordering: almost all pure entangled multipartite states are incomparable (i.e. LOCC transformations among them are almost never possible). In order to cope with this problem I will consider alternative resource theories in which I relax the class of LOCC to completely positive maps that do not create entanglement. I will show that this leads to a non-trivial theory: it induces a meaningful partial order since every state is transformable to more weakly entangled pure states. Moreover, I will also prove that it is possible to formulate a resource theory of multipartite entanglement which is not only non-trivial but has a unique maximally entangled state: the generalized GHZ state, which can be transformed to any other state by the allowed free operations.**Alvaro del Pino**(Universiteit Utrecht, The Netherlands)

**Bracket-generating integration**

In Control Theory, the Filippov-Wazewski relaxation theorem states that solutions of a differential relation are dense in the space of solutions of the corresponding convexified differential relation. In Geometry/Topology of PDEs, a similar idea appears in the work of J. Nash on C^1-isometric embeddings, which was later developed by M. Gromov into a extremely general method, called convex integration, to find solutions of ample partial differential relations.

I will review these ideas during my talk. I will then discuss some work in progress, joint with F.J. Martínez Aguinaga, in which we prove an analogous result for partial differential relations which are not ample but whose complement is bracket-generating. The proof borrows from the work of Gromov, as well as from classic ideas related to the Chow-Rashevskii theorem.

**José Figueroa-O’Farrill**(University of Edinburgh, UK)

**Homogeneous kinematical spacetimes: their geometry and symmetries**

Over 50 years ago, Bacry and Lévy-Leblond introduced the notion of a kinematical Lie algebra in four-dimensional spacetime and gave a classification. Some of their conditions were relaxed by Bacry and Nuyts twenty years later arriving at a classification of kinematical Lie algebras. They observed that each such Lie algebra acts transitively on some four-dimensional homogeneous spacetime and arrived at 11 classes of kinematical spacetimes. Recently with Stefan Prohazka we have refined and extended this pioneering work to arrive at a classification of simply-connected homogeneous kinematical spacetimes in arbitrary dimension and, in collaboration with my student Ross Grassie, we have also studied the geometry and symmetries of these spacetimes. In this talk I will summarise the highlights of these results.

**Andrea Fuster**(Eindhoven University of Technology, The Netherlands)

**Finsler gravity: what, why and how.**

In this talk I present the main ideas behind Finsler extensions of general relativity, known as Finsler gravity. I also give concrete examples of Finsler spacetimes which are exact solutions of the field equations in vacuum. Finally I comment on the (possible) physical relevance of such solutions.

**Bas Janssens**(Delft University, The Netherlands)

**Central extensions of hamiltonian and volume preserving diffeomorphisms**

The problem of classifying central extensions of a (possibly infinite dimensional) Lie group G can be split in two parts. Typically, one first solves the infinitesimal problem using Lie algebra cohomology. Having done this, one determineswhich of these infinitesimal extensions integrate to the group level. We report on recent progress on the first (infinitesimal) problem for the group of hamiltonian diffeomorphisms, and on the second (integration) problem for the group of exact volume preserving diffeomorphisms. For closed surfaces, where a symplectic form is the same as a volume form, these two results interact in a somehat unexpected manner. Based on joint work with C. Vizman, K.-H. Neeb and T. Diez.

**Franco Magri**(Università degli Studi di Milano-Bicocca, Italy)

**The Kowalewski's top revisited**

In this talk I will try to review a classical result by Kowalewski from a modern geometrical perspective. The study of the motion of the Kowalewski's top is, indeed, a beautiful problem where ideas of mechanics, algebraic geometry, and differential geometry nicely merge together.

**Madeleine Jotz Lean**(Georg-August-Universität Göttingen, Germany)

**Characteristic classes, ideals and representations up to homotopy.**

I will define connections up to homotopy of Lie algebroids on graded vector bundles, and show how the graded trace of the powers of their curvature induces characteristic classes of graded vector bundles. Using these, one can prove obstructions to the existence of a representation up to homotopy on a graded vector bundle. Then I will prove Bott's vanishing theorem in this more general setting and explain how it induces obstructions to the existence of ideals in Lie algebroids.

**Laura Schaposnik**(University of Illionis at Chicago, USA) - CANCELLED

**Giorgio Tondo**(Università di Trieste, Italy)

**Haantjes Geometry and Integrable Systems**

The notion of Nijenhuis torsion and the correspondent geometry has played a crucial role in the realm of integrable systems, since the introduction of recursion operators for soliton equations, by F. Magri in 1979. Very recently, Magri has noticed that the notion of Haantjes torsion is relevant in this realm and also in the framework of Frobenius manifolds, Veselov systems and topological field theories.

I will show that, in the context of Haantjes geometry, a central role is played by Haantjes algebras which are associative algebras of operator fields over a differentiable manifold, with vanishing Haantjes torsion. They describe in a unified way both integrable and superintegrable systems, the difference between them consisting in abelian or non abelian algebras of Haantjes operators. In both cases, when the Haantjes algebra associated with an integrable system is semisimple (diagonalizable), it can be solved by separation of variables.

Finally, I will present the new notion of Haantjes torsions of higher level m, which generalizes those of Nijenhuis and Haantjes torsion corresponding to m=1 and m=2, respectively. Such a notion will be useful in the theory of normal forms (diagonal, Jordan, triangular) of operator fields and can provide new insight in the theory of hydrodynamic type systems.