Testbrett

Ricci curvature is ubiquitous in mathematics: it appears in Hamilton's Ricci flow (a key tool in Perelman's resolution of the Poincaré conjecture), as well as in Einstein's equations of general relativity. Understanding its interplay with the global shape of Riemannian manifolds has been one of the key broad themes in geometric analysis since its early developments. While this interplay is well understood for manifolds with dimensions less than or equal to 3, several questions remain in dimension 4. After a gentle introduction to Ricci curvature, I will discuss joint work with Elia Bruè and Alessandro Pigati, in which we prove that any Riemannian 4-manifold with nonnegative Ricci curvature and Euclidean volume growth looks like a cone over a spherical space form at infinity. I will provide all the background needed for the precise statement, explain in which sense it is optimal, and explain why one might expect it to be true.

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We consider suspensions of many small rigid particles in a fluid. The theoretical study of the effective rheology, i.e. fluid properties, of such complex flows dates back to Einstein's doctoral research, where he predicted an effective increase of the fluid viscosity due to the presence of the particle. Despite its relevance for many applications and a vast physics and engineering literature on the topic, a mathematical rigorous derivation of effective rheology has been elusive until recent years. In this talk, I give an overview of results that not only justify Einstein's formal calculations but go far beyond. In particular, fascinating non-Newtonian effects will be presented.

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