Valid Until: 2027-04-30 23:59
Abstract: A complete Riemannian n-manifold M is called $\epsilon$-collapsed, if every unit ball in M has a volume less than \epsilon (while often a bound on `curvature' must be imposed to prevent a rescaling of metric). In 1978, Gromov classified `almost flat manifolds' (or the `maximally collapsed manifolds' with sectional curvature bounded in absolute value by one and small diameter) ; a bounded normal covering space of M is diffeomorphic to the quotient of a simply connected nilpotent Lie group modulo a manifold up to a co-compact lattice. This result has been a corner stone in the collapsing theory of Cheeger-Fukaya-Gromov in 90's that there is a nilpotent structure on any $\epsilon$-collapsed manifold with bounded sectional curvature, and this theory has found important applications in Metric Riemannian geometry.
We will survey some recent development in generalizing the collapsing theory to $\epsilon$-collapsed manifolds of Ricci curvature bounded below and the (incomplete) universal cover of every unit ball in M is not collapsed. The study of these collapsed manifolds is partially fuelled by many constructions of collapsed Calabi-Yau metrics using certain underlying singular nilpotent fibrations.
Bio: Xiaochun Rong is a distinguished professor at Rutgers University. He received his undergraduate and master's degrees from Capital Normal University (1978-1984), and his Ph.D. from the State University of New York at Stony Brook in 1990. After graduation, he was a Ritt assistant professor at Columbia University and an assistant professor at the University of Chicago. Then he became a tenured associate professor (1996) and professor (2002) at Rutgers. Professor Rong's research fields are Differential Geometry and Metric Riemannian Geometry. He has made several fundamental contributions to the convergence and collapse theory and their applications, the geometry and topology of a positively curved manifold, and an Alexandrov space. He has published over 50 papers in internationally renowned journals such as Adv. Math., Amer. J. Math., Ann. of Math, Duke Math., GAFA., Invent. Math., J. Diff. Geom, etc.
Professor Rong was rewarded the Sloan Research Fellowship in 1996 and was an invited speaker at the ICM 2002 in Beijing. He was elected a fellow of the American Mathematical Society in 2017.