From tuesday 15 to thursday 17, there will be two sessions a day; one in the morning (9:30-11:00) and one in the afternoon (14:00-15:30). It is not possible to attend to both lectures.

The two lectures are :

Gauduchon metrics: Existence and applications, by Nicolina Istrati

Summary: Gauduchon metrics are certain canonical representatives of a conformal structure. On a compact conformal manifold of dimension at least 3, they always exist and are unique up to constant factors, once one fixes a Weyl connection. Although they stem from conformal geometry, Gauduchon metrics are particularly interesting on complex manifolds. After a short introduction to conformal geometry and Weyl connections, we will define Gauduchon metrics and prove their existence. Then, after a short introduction to complex geometry, we will revisit Gauduchon's condition in this setting, show that it allows one to define the degree of a holomorphic line bundle and finish with a less formal discussion of some applications.

An introduction to analytic combinatorics, by Paul Thévenin

Summary: Analytic combinatorics is a recent topic whose aim is the asymptotic enumeration of discrete combinatorial structures of large size n (trees with n vertices, permutations on a set of n elements, graphs, …) using analytic methods. Specifically, let a(n) be the number of objects of size n that we want to study. Then, properties of the associated generating function F(x) = \sum_{n \geq 0} a(n) x^n provide information on the asymptotic behaviour of the sequence a(n). This allows us to transform a combinatorial problem (enumeration of structures) into an analytic one (usually, the study of a differential equation), using a now well-understood dictionary. The goal of this course is to introduce the main tools of analytic combinatorics, and to rigorously establish the most important results of this theory which is at the boundary between combinatorics, probability, real and complex analysis. I will mainly follow the seminal book of Flajolet and Sedgewick on this topic.