Séminaires doctorant en 2026

In this talk, I will discuss complex dynamics in dimension 1. After briefly reviewing Fatou-Julia theory, I will focus on family behavior. I will discuss bifurcation through the example of the Mandelbrot set. Then I will state and explain McMullen's classification theorem for stable families. Depending on the time remaining, I will present an arithmetic proof of this theorem by Z. Ji and J. Xie in 2023.

After a brief introduction to Fourier and Laplace transform theory, we show how the Taylor expansion of the dispersion relation provides information about the direction of propagation and the spreading of initially spatially localized waves. We then discuss a domain decomposition algorithm of the linear BBM equation and explain how the dispersion relation appears in the optimization of its convergence speed.

Dyer groups are a class that generalizes well-known families of groups such as Coxeter groups and RAAGs. There are some properties of these groups that extend naturally to the wider class, and one can use similar techniques to study them. In this talk, we will first introduce all these families of groups. Then, following previous work by Krammer in Coxeter groups and by Paris in Artin groups, we will construct a graph that gives an algorithm which decides whether two parabolic subgroups of a Dyer group are conjugated. This is a joint work with María Cumplido and Mireille Soergel.

The symmetric group appears in the Schur–Weyl duality describing the centralisers of tensor powers of the vector representation of the linear group. We would like to generalize this result to a new algebra called the fused permutations algebra. This last one was introduced recently for this purpose.
The first main goal will be to give an algebraic presentation of the fused permutations algebra, for a particular case, and a canonical basis. In particular, we prove that the fused permutations algebra is a quotient of the degenerate cyclotomic affine Hecke algebra, and we also describe a basis of this latter algebra combinatorially in terms of signed permutations with avoiding patterns.
The second main purpose, which comes from the first one, is the study of primitive idempotents of the cylotomic degenerate affine Hecke algebra. More precisely, we give a decomposition of these in a certain basis indexed by the elements of the Coxeter group of type B.