14th International Summer School on Geometry, Mechanics and Control

"Optimal control and the Maximum Principle", by Ravi Banavar (Indian Institute of Technology en Bombay, India)

 

Lec 1: A quick summary of the Calculus of Variations and introduction to some basic properties of convex sets, convex cones and separating hyperplanes

Lec 2: The free final time optimal control problem and the statement of the Pontryagin Maximum Principle (PMP). Start the proof: Temporal and Spatial perturbations

Lec 3: The propagation of the perturbed trajectory, convexity arguments, the adjoint system and the separating hyperplane

Lec 4: Completion of the proof and two examples: the minimum time problem for a translating mass, and the linear-quadratic regulator problem.

Lec 5: From the continuous time to discrete time: The Boltyanski Maximum Principle for discrete time systems; incorporation of constraints.

“Topics on spectral geometry”, by Alberto Enciso and Álvaro Romaniega (ICMAT-CSIC)

 

In this course we will discuss several results about nodal sets of eigenfunctions of the Laplacian on a Riemannian manifold. We will start from scratch an eventually present asymptotic results for random eigenfunctions. Throughout, we will emphasise the key role that solutions to the Helmholtz equation play in this context.
 

“Introduction to Lie Systems with Compatible Geometric Structures”, by Javier de Lucas (University of Warsaw)

 

This course surveys some of the most relevant geometric structures appearing in modern differential geometric theories: Poisson, symplectic, Dirac, Jacobi, multisymplectic, and k-symplectic manifolds. Next, we introduce the notion of Lie system, i.e. a nonautonomous system of first-order differential equations whose general solution can be described as a function, the superposition rule, of a particular generic family of particular solutions and some constants. The use of geometric structures for the calculation of superposition rules and constants of motion for Lie systems is to be analysed. Applications of the theory to systems of ordinary and partial differential equations of physical and mathematical relevance will be studied. Among other applications, we plan to look into nonautonomous frequency Smorodinsky-Winternitz oscillators, types of diffusion equations, and Backlund transformations for sine-Gordon equations.

 

“Hopf algebras, quantum groups and non-commutative geometry”, by Anna Pachol (Queen Mary University of London)

The main idea behind the noncommutative geometry is to "algebralize" geometric notions and then generalize them to noncommutative algebras. This way noncommutative geometry offers a generalised notion of the geometry. Quantum groups or Hopf algebras play the role of 'group objects' in noncommutative geometry and they provide a 'quantum groups' approach to the development of the theory much as Lie groups do in differential geometry. The term "quantum group" first appeared in the theory of quantum integrable systems and later was formalized by V. Drinfeld and M. Jimbo as a particular class of Hopf algebras with connection to deformation theory (as deformation of universal enveloping Lie algebra). Such deformations are classified in terms of classical r-matrix satisfying the classical Yang-Baxter equation.

We will start with the quantization of Poisson-Lie groups (i.e. Lie groups equipped with a Poisson structure) which provide a natural example of quantum groups. Other examples of Hopf algebras arising from Lie algebras through their universal enveloping algebras will also be discussed. The deformation procedure via Drinfel'd twist (2-cocycle) and some examples of deformed Hopf algebras will be presented together with the twisted deformation of the differential calculus.