Courses

Keywords: Markov processes, phylogenetic trees, applications to epidemiology

Phylodynamics is the study of processes that give rise to phylogenetic trees. This is a rapidly growing field fueled by advancements in genomic surveillance and computation efficiency; currently the big limitation of the field is a shortage in expertise and training. In this mini-course we will introduce the mathematical concepts involved in phylodynamics, including stochastic processes and Markov bridges. We will also work on simplified examples of applying phylodynamics to infer the parameters of a disease spreading through a population.

Keywords: Temporal difference learning, non-convex optimization, sample complexity

Root systems were introduced by Wilhelm Killing in 1889 as a tool for classifying Lie algebras. Since then they have become ubiquitous, appearing in various contexts in seemingly unrelated areas of Mathematics. In this mini course I will introduce and study root systems and will define a generalization of root systems. I will explain how this more general structure applies to various problems, e.g. hyperplane arrangements. Finally, I will introduce and study inversion sets of roots.

The course will assume reasonable knowledge of Linear algebra and very rudimentary knowledge of Group theory.

Keywords: To follow

Given a group G acting on a space X, one often wonders what kind of algebraic properties of the group can be deduced from the action. In another direction, one may also want to classify (or completely understand) certain families of actions, in the sense of trying to understand the dynamics of such actions. Answers to these wide ranging questions may take many forms, depending on the type of actions one consider and the space X, and leads to several distinct mathematical fields.

In this course, we will focus on actions on simple spaces, starting with X being the real line, and then considering X to be a bifoliated plane. Eventually, we will restrict to a class of actions called Anosov-like, which naturally appear when studying certain interesting type of flows on 3-manifolds, and we will cover some of the theory of such actions. In particular, we'll describe examples of these actions, give examples of algebraic properties one may deduce, and describe cases where one can obtain a classification.