Speakers

The 14th International Summer School on Geometry, Mechanics and Control will consist on three courses and a special session focused on geometric control. Additionally, there will be a number of short talks and a poster session by young researchers.

Courses

"Nodal sets of high-energy eigenfunctions of Schrödinger operators", by Alberto Enciso and Alba García-Ruiz (ICMAT-CSIC)

In this course we will study geometric properties of the nodal (i.e., zero) set of eigenfunctions of Schrödinger operators. As we will see, nodal sets of low-lying eigenfunctions turn out to exhibit a quite flexible behavior, while their high-energy asymptotic properties are considerably more rigid. We will explore these questions, which involve an interaction between harmonic analysis on the one hand, and geometric and dynamical considerations on the other. Throughout, connections with geometric aspects of Hamiltonian mechanics and with the fundamental equations of Physics will be emphasized.

"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 (University of South-Eastern Norway)

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.

Mini-course

"Geometric methods in optimal control", by David Martín de Diego (ICMAT-CSIC)

The purpose of this talk is to show how to address optimal control problems using modern techniques of geometric mechanics and how to build numerical methods preserving some of the geometric invariants of a given optimal control problem.

Short talks

Monday 4

  • 13:00-13:20 - Miguel Á. Berbel López (Universidad Complutense de Madrid), “Poisson–Poincaré reduction for field theories”

  • 13:20-13:40 - Anna-Katharina Hirmer (Friedrich-Alexander Universität Erlangen-Nürnberg), “Generalised Kitaev models from Hopf monoids: topological invariance”

Tuesday 5

  • 13:00-13:20 - Asier López-Gordón (ICMAT-CSIC, Madrid), “Routh reduction for hybrid forced systems”

  • 13:20-13:40 - Clarice Netto (IMPA), “Extending Poisson-Nijenhuis to Dirac-Nijenhuis structures”

  • 16:00-16:20 - David Martínez Crespo (Universidad de Zaragoza), “Hamiltonian Quantum Field Theory on a Cauchy Hypersurface”

  • 16:20-16:40 - Alberto José Balseyro Sebastián (University of Salamanca), “Solitary waves in massive non-linear S^1xS^1-Sigma models”

  • 17:30-17:50 - David Miguélez Caballero (Universidad de Valladolid), “Interaction between shape modes in a two-component scalar field theory”

  • 17:50-18:10 - Sergio Navarro Obregón (University of Valladolid), “Low energy dynamics of vortices with excited internal modes”

  • 18:10-18:30 - Carlos San Millán Carpintero (Universidad de Valladolid), “Supersymmetric partners of the One-Dimensional Infinite Square well Hamiltonian: Special Cases”

Thursday 7

  • 13:00-13:20 - Daniel Reyes Nozaleda (Universidad Complutense de Madrid), “Generalized Nijenhuis torsions and block-diagonalization of operator fields”

  • 13:20-13:40 - Bartosz Maciej Zawora (University of Warsaw), “An energy-momentum method on cosymplectic manifolds”

  • 16:00-16:20 - Sinuhe Perea (King's College London), “From Berry phase to quantum skyrmions: a geometric loop tour in Physics”

  • 16:20-16:40 - Martina Adamo (Universidad de Burgos), “Cosmological models evolving through the Big Bang”

  • 16:40-17:00 - Giuseppe Fabiano (University of Naples "Federico II"), “Modelling transverse relative locality at the Planck Scale”

  • 17:30-17:50 - Maria Grazia Di Luca (Scuola Superiore Meridionale), “ k-braided non-commutative field theory”

  • 17:50-18:10 - Diego Fernández-Silvestre (Universidad de Burgos), “Quantum effects in black holes and accelerated boundary correspondences”

  • 18:10-18:30 - Domenico Frattulillo (Università di Napoli "Federico II"), “Planck scale deformed relativistic transformations in curved space-times”

Posters

  • Sergio de la Huerta Sainz (Universidad de Burgos), “Quantum Revivals in Curved Graphene Nanoflakes”

  • Laura Elena González Bravo (Universidad Carlos III de Madrid), “The Wigner-Weyl transform in Molecular Physics”

  • Xavier Rivas (UNIR), “Lie systems and reduction of multisymplectic structures"

  • Daniel Torres Moral (Universitat Politecnica de Catalunya), “Lagrangian-Hamiltonian formalism for time-dependent dissipative mechanical systems”