François Labourie
Université Côte d'Azur
August 21, 2025
Positivity, Monotonicity and flag manifolds.: In this talk, I will explain in a geometric and accessible way the notion of « positivity » on (generalised) flag manifolds as introduced by Guichard and Wienhard. In order to motivate my talk, I will start with two examples of phenomena related to positivity: monotone mappings from the circle to itself, as well as convex curves in the plane. These two examples share elementary well known properties that I will explain. I will then give a dynamical definition of generalised flag manifolds and positivity and then show how the properties above are described as properties of positive curves in these generalised flag manifolds. I will then explain what are the consequences for representations of surface groups.
Université Côte d'Azur
August 21, 2025
Positivity, Monotonicity and flag manifolds.: In this talk, I will explain in a geometric and accessible way the notion of « positivity » on (generalised) flag manifolds as introduced by Guichard and Wienhard. In order to motivate my talk, I will start with two examples of phenomena related to positivity: monotone mappings from the circle to itself, as well as convex curves in the plane. These two examples share elementary well known properties that I will explain. I will then give a dynamical definition of generalised flag manifolds and positivity and then show how the properties above are described as properties of positive curves in these generalised flag manifolds. I will then explain what are the consequences for representations of surface groups.