The special lectures are particularly designed for VSM students. They also include mini-courses and short lecture series, with or without ECTS points.
Mini-Course: Cross-diffusion systems (with Ayman Moussa, Sorbonne Université) – April 1 to April 5, 2019
Cross-diffusion systems were introduced 40 years ago as a tool of modeling in the context of population dynamics. At the mathematical level, these systems happen to be surprisingly hard to study ; global (weak) solutions for the most simple of those were (only) built in 2006 even though they have been studied since the eighties.
We will start this lecture with a short introduction to present the model of Shigesada-Kawasaki-Teramoto (SKT, the first instance of cross-diffusion in the literature) and present (without proof) the existing results and the open questions related to it. Then, in the first part of the lecture, we will focus on the Kolmogorov equation, studying existence and uniqueness with the help of the duality lemma. In the second part of the lecture we will establish a global existence result for SKT systems, using a generalization of the entropy introduced by Chen and Jüngel in 2006, the duality lemma and a specific approximation procedure.
Mini-Course: Evolutionary equations with singular nonlinear terms (with Giulio Schimperna, Università di Pavia) – March 25 to March 28, 2019
After recalling a number of basic notions from convex analysis (maximal monotone operators, subdifferentials), I will focus on some mathematical aspects of evolutionary partial differential equations characterized by the presence of singular (i.e., very fastly growing) terms. I will discuss existence of solutions, regularity properties, uniqueness (or non-uniqueness), and long-time behavior.
Moreover, I will try to illustrate the theory by applying the results to some specific problems related to physical models from the theory of phase transitions.
Mini-Course: Combinatoris, groups and algebraic varieties (with Fernando Rodriguez Villegas, ICTP Trieste) – November 5 to November 14, 2018
In this course we will discuss how techniques from enumerative combinatorics and the representation theory of finite groups can be used to study the geometry of certain algebraic varieties. Examples will include some character varieties as well as some related quiver varieties. We will introduce all the basic theory needed from scratch.
Professor Fernando Rodriguez Villegas is a Senior Research Scientist at the Abdus Salam International Centre for Theoretical Physics (ICTP) in Trieste, Italy. His research interests lies in number theory, in the study of L-functions (particularly those coming from hypergeometric motives), character varieties, combinatorics of quiver representation, etc.
Mini-Course: Spatial Discretizations of Generic Dynamical Systems (with Pierre-Antoine Guihéneuf, IMJ-PRG, Paris 6) – November 20 to November 24, 2017
Abstract: How is it possible to read dynamical properties of a system on numerical simulations? The computer working with finite numerical precision, it replaces the dynamics f by a spatial discretization f_N (where stands for the numerical accuracy, e.g. the number of binary digits). We will be interested in the dynamical behaviour of the finite maps f_N for a generic system f and N going to infinity (mainly among sets of homeomorphisms or C^1-diffeomorphisms). These lectures will be enlightened by numerical simulations, and will also be the occasion to understand some tools of generic dynamics.
Mini-Course: Topological Methods in Aperiodicity (with James J. Walton) – June 7 to June 9, 2017
Mini-Course: An Introduction to Noncommutative Geometry and Quantum Groups (with Réamonn Ó Buachalla) – May 29 to June 2, 2017
Mini-Course: A tale of two phenomena: Fixed Point Property and Minimal Dynamical Systems – May 16 to May 19, 2017
Mini-Course: Dynamics of the unimodal interval family (with Ana Anusic) – May 3 to May 5, 2017
Mini-Course: Permutation groups and transformation semi-groups (with Prof. Peter Cameron) – March 20 to March 24, 2017