October 7th, 2026

12:15 - Radu Ignat (IMT)

Amphi Schwartz, Institut de mathématiques

Symmetry of domain walls in some elliptic PDE systems

Domain walls are topological solitons that appear in many physical systems (phase transitions, condensed matter etc). They represent transition layers between two states corresponding to the wells of a certain potential function. We will start by presenting the domain walls in the Allen-Cahn model where the potential has a finite number of wells; in particular, we will study their symmetry and energy level. Next we study the case of Ginzburg-Landau type models where the potential has a continuum of wells. For well-posedness, the divergence-free constraint is imposed on the order parameter. In this case, we will study one-dimensional symmetry of domain walls as well as the nucleation of microstructures according to the shape of the potential function.

October 21th, 2026

Benoît Bertrand (IMT)

Salle F. Pellos (1R2-207), Institut de mathématiques

A glance at Hilbert 16th problem

A real plane algebraic curve of degree d is the cancellation locus of a degree d real polynomial.

This set of real zeroes can have many connected components (but not too many, by Harnack's theorem). The first part of Hilbert 16th problem is about the possible relative positions of these components. In general this is a wide-open problem. The solution is only completely know up to degree 7. I will explain some properties of these curves, set up the problem, and explain methods that lead to the classification of the curves up homeomorphism of the ambient plane in very small degrees.

https://indico.math.cnrs.fr/event/17377/