AGAPI Research group (CSIC-UAM-UPM)



Wednesday March 16th (2022)

15:30 Antoine Prouff (Univ. Paris-Saclay). Damped waves in the Euclidean space
Adding a damping term to the wave equation produces a decay of the energy of solutions. The question of stabilization of waves consists in finding necessary and/or sufficient conditions on the damping coefficient to ensure that the trend to equilibrium is uniform with respect to the intial data. In a series of seminal works, Rauch and Taylor in the 70s, and later Bardos, Lebeau and Rauch, solved this problem in the setting of compact Riemannian manifolds. Their results reveal the importance of geometric optics (or geodesic flow) on the manifold. In this talk, we will recall briefly the framework of semiclassical analysis, and explain how it allows to tackle the problem of stabilization using the concept of quantum-classical correspondence. Then we will consider the case of the Euclidean space, that is not well-understood yet, due to new difficulties related to the unboundedness of the domain. We will specifically discuss the interesting situation where the geometric optics is distorted by the presence of a confining potential.
16:30. Alba D. García-Ruiz (ICMAT) High-energy eigenfunctions and inverse localization
A well-known link between monochromatic waves (i.e. solutions to the so-called Helmholtz equation \(\Delta v+v=0\)) and eigenfunctions on a Riemannian manifold \((M,g)\), \(\Delta_g u+\lambda^2 u=0\), is that the local behaviour of a sequence of high-energy eigenfunctions (say, \(\lambda^2\rightarrow\infty\)) defines a bounded monochromatic wave, after suitable rescalings that depend on the eigenvalue. Conversely, the local behaviour of a solution to Helmholtz can be realized by an approximate eigenfunction (or quasimode) of any large enough energy, on scales determined by this energy. A powerful refinement of the latter fact is what we call the inverse localization principle: if, roughly speaking, the degeneracy of the high-energy eigenvalues is large enough, one can replace the quasimodes by \emph{bona fide} eigenfunctions. In this talk we prove precise versions of this principle for the harmonic oscillator potential (in \(\mathbb{R}^d\) and \(\mathcal{H}^d(\kappa)\), the hyperbolic space of constant curvature \(-\kappa^2\)) and for the Laplacian on the sphere and certain flat tori. Finally, we mention some key ideas behind the localization principle in order to develop a systematic theory, e.g. the underlying symmetry hypothesis present in all the contexts mentioned above.
17:40 Juan Viu-Sos (Univ. Politécnica de Madrid) On the embedded topology and combinatorics of projective line arrangements
A \emph{line arrangement} \(\mathcal{A}=\{L_0,\ldots,L_n\}\) is a collection of finitely many lines in \(\mathbb{C}\mathbb{P}^2\), whose \emph{combinatorics} is described by the \emph{incidence graph} \(\Gamma_\mathcal{A}\) between lines and points. A classical topic for line arrangements is the study of the influence of \(\Gamma_\mathcal{A}\) on the topology of the embedding \((\mathbb{C}\mathbb{P}^2,\mathcal{A})\). A pair of combinatorially equivalent arrangements with different topological type is called a \emph{Zariski pair}. In this talk, we make a historical introduction about Zariski pairs, the (few) examples which were known until now for line arrangements and some related topological invariants (e.g.~the fundamental group of the complement). Also, we present a method to construct Zariski pairs based on weighted configurations of points in \(\mathbb{R}\mathbb{P}^2\), jointly developed with B.~Guerville-Ballé. Combining our result with the use of {\sc GeoGebra}, we will construct a new Zariski pair of 13 lines with many interesting properties.

Venue ^

ETSIN Aula 5, ETSI Navales, Avda. de la Memoria 4, Madrid.

Financiado por ^