Università Bocconi, Milano

Title: The fundamental groups of manifolds with nonnegative Ricci curvature

Abstract: In 1968, Milnor proposed a conjecture stating that Riemannian manifolds with nonnegative Ricci curvature have finitely generated fundamental groups.

The first part of the course will focus on comprehending this conjecture and discussing the progress made in addressing it throughout the years.

In the subsequent part, we will discuss recent work with Naber and Semola, where we provide a counterexample to Milnor's conjecture. More specifically, we construct a seven-dimensional manifold with nonnegative Ricci curvature whose fundamental group is isomorphic to Q/Z. The key ideas behind the construction will be explained, and the resulting manifold's geometry will be described in detail.

Université  Côte d'Azur 

Title: Introduction to semialgebraic and subanalytic geometry.

Abstract: The goal of this course is to give an introduction to semialgebraic and subanalytic sets and mappings with emphasis on the properties used in various areas of differential geometry and analysis. In particular we cover such subjects as Łojasiewicz Inequalities, decompositions and stratifications. One lecture will be devoted to the metric properties of these sets including the L-regular decomposition, subanalytic preparation theorem and Lipschitz stratification. We also discuss the integration on subanalytic sets, local densities, and the dependence of integrals on parameters in families.