Encounter with a geometer. II (Q2756726)

This is the second part [for Part I, see ibid. 47, No. 2, 183-194 (2000; Zbl 1030.53002)] of an expository article devoted to M. Gromov's work and its influence in contemporary mathematics. The author discusses M. Gromov's revolutionary results and ideas in Riemannian geometry. Topics include topological restrictions for positively curved manifolds (a universal upper bound for the Betti numbers), negatively curved manifolds (finiteness results, pinching, arithmeticity), positive scalar curvature (\(K\)-area), Gromov's program of synthetic Riemannian geometry, manifolds with Ricci curvature bounded from below, collapse, the spectrum of Riemannian manifolds, periodic geodesics, Kähler geometry. A substantial part of the article is devoted to Gromov's program of building a ``friendly environment for treating asymptotics of many interesting spaces of configurations and maps''. This is the space of metric-measure spaces, mm-spaces, endowed with the Gromov distance. The key notion of observable diameter and its estimates based on a concentration phenomenon are discussed. The article is concluded by quoting M. Gromov on his mathematical perception and process of discovery.