Florio M. Ciaglia (Max-Planck-Institut, Leipzig)
From the Jordan product to Riemannian geometries
Friday the 13th of December, 2019, 13:00, ICMAT, Aula Gris I
The Jordan product on the self-adjoint part of a finite-dimensional C*-algebra A is shown to give rise to Riemannian metric tensors on suitable manifolds of states on the algebra. In particular, this construction allows to look at the Fisher-Rao metric tensor on probability distributions, at the Fubini-Study metric tensor on pure quantum states, and at the Helstrom metric tensors on faithful quantum states as different instances of the ''same conceptual entity''. If time allows, an alternative derivation of these Riemannian metric tensors in terms of the GNS construction will be presented.
José Polo (UCM)
Groupoids and inverse semigroups
Monday the 2nd of December, 2019, 15:30, UC3M, Seminar Room 2.2D08
We aim to introduce briefly the concepts of groupoid and inverse semigroup, motivating and showing the relation between them by means of some examples. We will then turn to the construction of groupoid representations as well as their associated C*-algebras, illustrating the construction with a class of examples that includes Cuntz-Krieger algebras.
Olaf Post (Universität Trier)
Spectral gaps, discrete magnetic Laplacians and spectral ordering
Friday the 15th of November, 2019, 12:30, ICMAT, Aula Gris 1
In this talk, we localise the spectrum of a discrete magnetic Laplacian on a finite graph using techniques similar to the Dirichlet-Neumann-bracketing for continuous problems. As application we localise the spectrum of periodic Laplacians using the fact that the fibre operators from Floquet Bloch theory can be seen as magnetic Laplacians. Finally, we use the bracketing ideas to order spectra of different graphs, and show how certain graph manipulations change the spectrum.
David Krejcirik (Czech Technical University, Prague)
Spectral geometry of quantum waveguides
Friday the 13th of September, 2019, 15:00, ICMAT, Aula Naranja
We shall make an overview of the interplay between the geometry of tubular neighbourhoods of Riemannian manifold and the spectrum of the associated Dirichlet Laplacian. An emphasis will be put on the existence of curvature-induced eigenvalues in bent tubes and Hardy-type inequalities in twisted tubes of non-circular cross-section. Consequences of the results for physical systems modelled by the Schroedinger or heat equations will be discussed.