MATJ5128 MA3: Curvature Bounds and Nilpotent Structures (JSS35) (2 cr)

A central theme in Riemannian geometry is the study of the interplay between curvature and topology. Particularly relevant is the investigation of algebraic restrictions on the fundamental groups of manifolds satisfying different curvature bounds. Starting from Gromov’s celebrated almost flat theorem for manifolds with bounded sectional curvature, and the Fukaya–Yamaguchi almost nilpotency theorem for spaces with lower sectional curvature bounds, it has become clear that nilpotent structures play a fundamental role in this context. The aim of this course is to review these classical results and key examples, and then to focus on more recent developments involving Ricci curvature bounds, including progress and perspectives related to the Milnor conjecture.     

After completing the course, the student will:

- Understand the relationship between curvature bounds and algebraic properties of fundamental groups, with particular emphasis on nilpotent and abelian structures.

- Be familiar with foundational results such as Gromov’s almost flat theorem and the Fukaya–Yamaguchi almost nilpotency theorem, and understand their role within the broader landscape of Riemannian geometry.

- Gain exposure to modern techniques, current research directions, and open problems in the study of Ricci curvature and topology, and develop the background needed to approach contemporary research literature in the field.

Basic Riemannian geometry (complete manifolds, connections, geodesics, curvature) and algebraic topology (fundamental groups and covering spaces).