The annual meeting of Physics Lab (LPT) and Maths Lab (IMT)
Since 2009, the annual meeting of the Theoretical Physics Lab and Mathematics Lab gathers physicists and mathematicians around the thematics of MathematicalPhysics and Theoretical Physics.
The program for the 2023 edition is as follows:

10h11h: Julien Royer  Local energy decay for the damped wave equation

11h12h: Clément Sire  Hilbert parameterfree interpolation kernel: exact results and application to the generation of reallife fish trajectories

12h13h30 lunch

13h3014h30: Anna Szczepanek  Invariant measures for quantum trajectories and dark subspaces (Physicoriented version)

14h3015h30: Luka Trifunovic  Fragility of spectral flow for topological phases in nonWignerDyson classes
You can find the abstracts below.
Local energy decay for the damped wave equation.
In this talk I will introduce the question of local energy decay for the damped wave equation in unbounded domains. We will discuss some general ideas and methods used in this kind of context, in particular to estimate resolvents in weighted spaces. I will also give some examples of results, with a damping localized in a bounded region or effective at infinity.
Fragility of spectral flow for topological phases in nonWignerDyson classes
Topological insulators and superconductors support extended surface states protected against the otherwise localizing effects of static disorder. Specifically, in the WignerDyson insulators belonging to the symmetry classes A, AI, and AII, a band of extended surface states is continuously connected to a likewise extended set of bulk states forming a ``bridge’’ between different surfaces via the mechanism of spectral flow. In this work we show that this principle becomes fragile in the majority of nonWignerDyson topological superconductors and chiral topological insulators. In these systems, there is precisely one point with granted extended states, the center of the band, E=0. Away from it, states are spatially localized, or can be made so by the addition of spatially local potentials. Considering the threedimensional insulator in class AIII and winding number ν=1 as a paradigmatic case study, we discuss the physical principles behind this phenomenon, and its methodological and applied consequences. In particular, we show that lowenergy Dirac approximations in the description of surface states can be treacherous in that they tend to conceal the localizability phenomenon. We also identify markers defined in terms of Berry curvature as measures for the degree of state localization in lattice models, and back our analytical predictions by extensive numerical simulations. A main conclusion of this work is that the surface phenomenology of nonWignerDyson topological insulators is a lot richer than that of their WignerDyson siblings, extreme limits being spectrum wide quantum critical delocalization of all states vs. full localization except at the E=0 critical point. As part of our study we identify possible experimental signatures distinguishing between these different alternatives in transport or tunnel spectroscopy.
Invariant measures for quantum trajectories and dark subspaces
Quantum trajectories are Markov chains modeling the evolution of a quantum system subject to repeated indirect measurements. The renowned purification theorem by Kümmerer and Maassen [1] says that, asymptotically, a quantum trajectory either reaches the set of pure states or it gets trapped in a set of `dark subspaces’, i.e., subspaces from which no information can be extracted, where it performs a random walk. It was shown in [2] that in the case of (irreducible) systems with no dark subspaces, so when purification is guaranteed, quantum trajectories admit a unique invariant probability measure on the set of pure states and the convergence towards this measure is geometric. This talk concerns the case of systems that do have dark subspaces. By using maxlikelihood estimation, we can show that the random walk between dark subspaces admits a unique invariant probability measure and that the convergence towards it is again geometric. We also discuss the problem of characterizing invariant measures for quantum trajectories inside dark subspaces, in particular we give some necessary conditions for uniqueness. Joint work with T. Benoist and C. Pellegrini.
Hilbert parameterfree interpolation kernel: exact results and application to the generation of reallife fish trajectories
Given a set of (training) data consisting of inputs and outputs, an interpolation scheme is a method to construct a function which exactly reproduces the training data and which hopefully provides a “good” prediction of the outputs for new (test) inputs not in the training dataset. I will give a very brief overview of interpolation kernel methods and will ultimately focus on the completely parameterfree Hilbert kernel, for which I have obtained rigorous results about its probabilistic convergence when the number of data grows (real theorems by a physicist!). Finally, I will give a practical application of the Hilbert kernel interpolation scheme as a generative model for realistic fish trajectories in groups of 2 and 5 fish.