There will be three 4-hour courses:  

• Tomoki Ohsawa (University of California, San Diego): Hamilton-Jacobi theory
  I will talk about Hamilton-Jacobi theory with a special emphasis on its relationship with geometric mechanics. Starting from the basic concepts in Hamilton-Jacobi theory, I will talk about the geometry associated with Hamilton-Jacobi theory and its applications. The first lecture gives some basic concepts such as Jacobi's solution as well as applications to integrable Hamiltonian systems. The second lecture is on the geometric interpretation of the Hamilton-Jacobi equation. The third lecture will focus the relationship with optimal control theory (and, time permitting, quantum mechanics). The fourth lecture gives the contact-geometric formulation of the Hamilton-Jacobi equation.

• Mario Sigalotti (INRIA, France): Quantum control
  In the first part of the course (~ 1 hour) we will present a panorama of some classes of control systems which are studied in the framework of quantum control. In particular, we will introduce the main features characterizing such systems: closed vs open, finite- vs infinite-dimensional, (bi)linear vs nonlinear. We will give some basic properties of the class of systems on which the rest of the course will focus on: closed, finite-dimensional, bilinear systems. The central part of the course (~ 2 hours) will be devoted to controllability and motion planning. We will present some general controllability results of finite-dimensional systems (Krener's and Chow's theorem, compatibility of Poisson stable vector fields) and apply them to the quantum setting. We will provide some explicit conditions for controllability and discuss some algorithms for motion planning. In the final part of the course (~1 hour) we will discuss some specific optimal control problems (mainly, time-minimal problems for 2- or 3-level quantum systems).

• Ivan Struchiner (Utrecht University, The Netherlands): Lie algebroids and groupoids
  These lectures are intended to be an introduction to Lie groupoids and Lie algebroids. The first lecture will be entirely dedicated to Lie groupoids. After defining the main concepts (such as Lie groupoids and Morita equivalence of Lie groupoids), several examples will be presented. The infinitesimal counterpart of a Lie groupoid, its Lie algebroid, will be introduced in the second lecture. I will show how to construct the Lie algebroid of a Lie groupoid, provide several examples and then discuss some aspects of the Lie theory for Lie groupoids/algebroids. The third lecture will be dedicated to cohomology theories naturally associated to Lie algebroids (and Lie groupoids). I will discuss the de Rham and deformation cohomology of a Lie algebroid. If time permits, I will discuss also the differentiable cohomology of a Lie groupoid. In the last lecture I will present a geometric proof of the linearization theorem for proper Lie groupoids, which is based on a deformation argument (in the spirit of Moser's trick).

The confirmed speakers for the Geometry & Physics day are:

• Camille Laurent-Gengoux (Paul Verlaine University-Metz, France): Global action-angle variables
  Local action angle variable always exist for non-commutative integrable
  system. Classes of cohomology, and, surprisingly, some non-linear equation,
  obstruct their global existence.
• Juan Carlos Marrero (University of La Laguna, Spain): Kinematic reduction and the Hamilton-Jacobi equation
  In this talk, I will present a close relationship between the classical Hamilton-
  Jacobi theory and the kinematic reduction of control systems by decoupling vector fields.
  The geometric interpretation of this relationship relies on new mathematical techniques for
  mechanics defined on a skew-symmetric algebroid. This geometric structure allows us to
  describe in a simplified way the mechanics of nonholonomic systems with both control and
  external forces.
• Edith Padrón (University of La Laguna, Spain): The contact cover of the spherical bundle of a fiberwise linear Poisson manifold
  If $A$ is a Lie algebroid over $M$ of rank $n$ and $g$ is a
  bundle metric on $A$, we may consider the corresponding spherical
  bundle $S^{n-1}(A^*)\to M$ of the linear Poisson manifold $A^*$. In
  this talk, we will describe a contact structure on the prolongation of
  $A$ with respect to this spherical bundle. This fact allows to prove
  that $S^{n-1}(A^*)$ is a Jacobi manifold. We will describe these
  structures in several examples of Lie algebroids.
• Mathieu Stienón (Penn State University, USA): From Atiyah classes to homotopy Leibniz algebras
• Wlodzimierz M. Tulczyjew (University of Warsaw, Poland): Hamilton principal form and holonomic constraints.
  Hamilton principal form is defined in general and
  presented inparticular to mechanical systems with holonomic
  constraints.
• Ping Xu (Pen State University, USA): Glanon groupoids
  We introduce the notion of Glanon groupoids, which are Lie groupoids
  equipped with multiplicative generalized complex structures.
  It combines symplectic groupoids, holomorphic Lie groupoids and
  holomorphic Poisson groupoids into a unified framework. Their
  infinitesimal, Glanon Lie algebroids are studied. We prove that
  there is a bijection between Glanon Lie algebroids and source-simply
  connected and source-connected Glanon groupoids. As a consequence,
  we recover various integration theorem and obtain the integration
  theorem for holomorphic Poisson groupoids.
  This is a joint work with Jotz, and Stienon .
• Marco Zambon (ICMAT and Universidad Autónoma de Madrid, Spain): Singular foliations and holonomy groupoids
  I will recall the notion of holonomy for a regular foliation on
  a smooth manifold, and the related notion of holonomy groupoid. I
  will then introduce singular foliations (the simplest example is given
  by a vector field, possibly with singularities) and sketch a
  construction of an associated holonomy groupoid by
  Skandalis-Androulidakis. The latter is in general not smooth. I will
  show that, however, under suitable conditions the restriction to leaves
  is smooth. The content of this talk is the first part of a project to
  find a normal form theorem for singular foliations, and is joint work
  with Androulidakis.

Short talks

• Letizia Brunetti: An Osserman-Type Condition In Lorentz Context
• Thiago Drummond: Right-invariant derivations on Lie groups
• Eduardo García-Toraño: Aspects of Routh Reduction
• Elisa Lavinia Guzmán Alonso: An involution map for the iterated jet of a surjective morphism of Lie algebroids
• Madeleine Jotz: Some multiplicative geometric structures on Lie groupoids and the corresponding infinitesimal data
• Derek Krepski: Verlinde-type formulas for SO(3) through geometric quantization
• Ignazio Lacirasella: On some reduced Poisson manifolds foliated by coadjoint orbits
• Hassan Najafi Alishah: A geometric approach to the KAM theorem
• Romero Solha: Almost homotopy operators in geometric quantisation of integrable systems
• Pietro Tortella: $\Lambda$-modules and holomorphic Lie algebroid connections
• Silvia Vilariño Fernández: k-symplectic manifolds and their Lagrangian submanifolds
• Goedele Waeyaert: Driven cofactor systems and Hamilton-Jacobi separability