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# Journée Mathematique Physique

23 mai 2018
IMB
Europe/Paris timezone

Programme de la journée :

• 13 H - L. Al Sheikh - Titre : Towards an approach to quasi-normal modes in optical systems: from complex to spectral analysis
•  Résumé

We discuss the first elements of an approach for studying qualitative and quantitative
aspects of Quasi-Normal Modes (QNMs) of a one-dimensional optical system. In a first
step we consider a permittivity depending on the frequency, without absorption, according
to the so-called Drude model. Linearizing the permittivity around the frequency, we get a
new wave equation with an effective potential. In a second step we overview the treatment
of QNMs of a 1-dimensional wave equation with a potential, and we discuss the resonant
expansion (in time) of a scattered field in terms of QNMs. Finally, we comment on a generalization
of this discussion for spaces with higher (odd) dimensions, in terms of results in spectral theory.

• 13 H30  G. Kulkarni - Titre Asymptotic analysis of the form-factors of the XXX chain

• Résumé : A quantum integrable system solvable by algebraic Bethe ansatz (ABA) admits an L-matrix and a R-matrix using satisfying the quantum Yang-Baxter equation using which we can construct a transfer matrix compatible with the Hamiltonian of the system and generating an infinite family of conserved charges through the trace identities. This algebraic construction permits us not only to obtain analytically the spectrum of the Hamiltonian but also gives us, in principle, the determinant representation for the overlaps of the eigenvector with an arbitary vector (Slavnov determinant) and the solutions to the quantum inverse scattering problem which relates the local observables to the global observables used in the ABA construction. These are the essential ingredients to compute the thermodynamic quantities such as the form-factors, which is defined as the overlaps of a local observable between the ground state vector of the system and an excited state vector. The exact computation of the form-factors enables the exact computations of the thermodynamic quantities such as the two-point correlation function, the dynamic structure factor which play important role in describing the physics of the model and since recently they have been probed experimentally. This talk concerns our ongoing work on the asymptotic computation of the longitudinal form-factors of the XXX chains in the thermodynamic limit from the purely ABA based method. Here we attempt to reproduce the previous results obtained by Bougourzi et al using the q-vertex operator algebras as the XXX limit of the massive XXZ form-factors. However in contrast to the previous approaches, our approach is more direct and can also be used to compute the form-factors of masless XXZ chains in more wider scenarios.

• 14h00: A. Kulkarni - Titre : Distinguishing modular categories by topological invariants

• Résumé : Given a finite group $G$ and a 3-cocycle $\omega \in H^3(G, \mathbb{C}^\times)$, it is possible to construct a pointed fusion category. These may be seen as toy models of fusion categories, which are tensor categories with some specific finiteness conditions. A construction known as the 'Drinfeld center' of a fusion category gives rise to a modular category. Classifying the Drinfeld centers upto equivalence amounts to classifying pointed fusion categories upto (categorical) Morita equivalence. I will report on recent progress in distinguishing non-Morita equivalent pointed fusion categories that could not be distinguished using only classical invariants of modular categories.

• 14h30-14h45: break

• 14h45: U. Muhammad - Titre : Nontrivial boundaries and blow-up investigation in the Elliptic-Hyperbolic Davey-Stewartson's System

• Résumé : We study the Cauchy problem of a particular Davey-Stewartson Equation (2+1 dimensional) generalizaing the 1+1 dimensional Nonlinear Schrödinger equation. The equation is solved for sufficiently small initial data and nontrivial boundary conditions which allows for localised travelling waves. We study blow-up of solutions for localised initial data.

• 15h15: O. Assainova - Titre : On semiclassical analysis for «dbar» problem.

• Résumé :

Abstract:  Integrable systems has been vastly studied in the XX century. Throughout this time a large number of the powerful methods were developed. Probably one of the most efficient is  the inverse scattering method.  We will briefly introduce the details and show how this method can be implemented  in case of the  two-space dimensional Davey-Stewartson system  and how the announced «dbar» problem appears in this context. It will be shown that semiclassical limit of the  problem can be used not only  as tool that provides some new  understanding and the perspectives of the problem but also as a algorithmic way of constructing  solutions.

•

15h45: N. Stoilov - Titre : Electric Impedance Tomography

• Résumé : Electric Impedance Tomography (EIT) is a medical imaging technique that uses the response to voltage difference applied outside the body to reconstruct tissue conductivity. As different organs have different impedance, this technique makes it possible to produce images of the body without exposing the patient to potentially harmful radiation.
In mathematical terms, EIT is as an inverse problem, whereby data inside a given domain is recovered from data on its boundary. In contrast with techniques like X-ray tomography (based on a linear problem), the particular inverse problem employed in EIT is non-linear - it reduces to a D-bar problem. Such problems also find application in the area of Integrable Systems, specifically in the 2+1 dimensional Davey - Stwartson equation.
I will discuss the design of numerical algorithms based on spectral collocation methods that address D-bar problems found both in integrable systems and medical imaging. Successfully implementing these methods in EIT
should allow us to achieve images with much higher resolutions at reduced post-processing times. Our approach is highly parallelisable, meaning it can be accomplished with graphical processing units (GPUs) for further efficiency gains without increasing the cost of the process. As we develop the technique, we hope that the speed and convenience of EIT, which employs a belt placed around the patient’s body, will make it a less intrusive and distressing form of medical imaging.

• 16h15-16h30: break

• 16h30: O. Lisovyy - Titre : Tau functions as Widom constants

• Résumé : I will explain how to assign a tau function to the Riemann-Hilbert problem set on a union of non-intersecting smooth closed curves with generic jump matrix. The main focus will be on the one-circle case, relevant to the analysis of Painlevé VI equation and its degenerations to Painlevé V and III. The tau functions in question will be defined as block Fredholm determinants of integral operators with integrable kernels.

Démarre 23 mai 2018 13:00
Finit 23 mai 2018 18:00
Europe/Paris
IMB
318 Aile A