Tejeddine Ghoul (New York University Abu Dhabi):

   Stable self-similar blowup for 3-D incompressible Euler equations (Cours Avancé).

Abstract: Recently Elgindi proved the existence of a continuum of C^{1,alpha} self-similar solutions to 3D axisymmetric Euler  equations with vanishing swirl. The aim of the talk will be to show that those solutions are stable under perturbations with swirl. This is the first result proving the existence of finite energy blowing up solutions for 3D Euler equations . I will in a first part of my presentation talk about recent result we obtained on one dimensional models for which we proved the stability of smooth and C^{1,alpha} self-similar solutions. In a second part I will explain how those one dimensional models helped us understanding the full 3D Euler. And in a last part I will explain how those results help understanding the millennium problem on 3D Navier-Stokes.

Hichem Ben El Mechaiekh (Brock University, Canada)

   On a class of set-valued maps and its role in nonlinear functional analysis.

Abstract: The talk will discuss the nature and properties of an important class of point-to-set maps and its fundamental role in nonlinear functional analysis. The class consists of maps that are graphically approximable by continuous mappings. It includes set-valued maps with convex or non-convex values and enjoys remarkable stability properties. It plays a fundamental role in topological fixed point theory and in the solvability of nonlinear problems. Applications to the viability of dynamical systems, optimization and Mathematical Economics will be briefly discussed, time permitting.

Othman Echi (King Fahd University)

   On the language of primitive words

Abstract: Let p ,q  be two distinct primitive words. According to Lentin-Sch\"utzenberger,  the language p^+q^+ contains at most one non-primitive word and if  pq^m  is not primitive, then $m\leq \dfrac{2\mid p\mid}{\mid q\mid}+3$. In this talk, we review some well-known results about primitive words and give a sharper upper bound, namely, $m\leq \lfloor\dfrac{\mid p\mid-2}{\mid q\mid}+2\rfloor,$ where $\lfloor x\rfloor$ stands for the floor of $x$.

Sadok Kallel (American University of Sharjah et Université de Lille)

   On applications of algebraic topology to problems in geometric analysis.

Abstract: On explicitera d'une facon simplifiée deux ou trois applications de la topologie algébrique dans la résolution de problèmes liés à l'analyse géométrique. L'idée est de familiariser l'audience avec quelques constructions classiques mais utiles de la topologie.

Mohamed Kamel Riahi (Khalifa University, Abu Dhabi)

A robust inversion method for quantitative 3-D shape reconstruction from coaxial eddy-current measurements.

Abstract: This work is motivated by the monitoring of conductive clogging deposits in the steam generator at the level of support plates. One would like to use monoaxial coils measurements to obtain estimates on the clogging volume. We propose a 3D shape optimization technique based on a simplified parametrization of the geometry adapted to the measurement nature and resolution. The direct problem is modeled by the eddy current approximation of time-harmonic Maxwell's equations in the low-frequency regime. A potential formulation is adopted in order to easily handle the complex topology of the industrial problem setting. We first characterize the shape derivatives of the deposit impedance signal using an adjoint field technique. For the inversion procedure, the direct and adjoint problems have to be solved for each coil vertical position which is excessively time and memory consuming. To overcome this difficulty, we propose and discuss a steepest descent method based on a fixed and invariant triangulation. Numerical experiments are presented to illustrate the convergence and efficiency of the method.