**Title:** Tamed and wild behaviors of canonical Riemannian metrics

**Abstract:** Global Riemannian geometry studies the interaction between curvature and topology. For example, starting in the 1950s, a series of sphere theorems were established in the literature, including Rauch's Sphere Theorem (1951), Berger-Klingenberg's Quarter-Pinched Sphere Theorem (1960), Grove-Shiohama's Diameter Sphere Theorem (1977), etc. A common feature of all these theorems is the pursuit of a classification of the (differential) topological structure of a manifold given certain curvature restrictions. Starting in the 1970s, Cheeger and Gromov established several fundamental results on the diffeomorphism finiteness and topological boundedness, providing another framework that is much more general than studying a particular model geometry such as the round sphere. Roughly speaking, the question of the uniform tameness and development of singularities of a class of manifolds that satisfy certain "loose" curvature conditions has become a rather active topic. This folklore theme has inspired both substantial developments in various research areas of differential geometry and has significant impacts on many other subjects such as algebraic geometry, analysis of nonlinear PDEs, and mathematical physics.

The main geometric object in this talk is the Einstein metric. By definition, an Einstein metric satisfies a system of nonlinear equations that can be viewed as a certain nonlinear analogue of harmonic equations. We will explore the rich geometric/topological structures that Einstein metrics can lead to, as well as their applications in understanding the metric geometry and global topology of underlying spaces. We will particularly discuss which kinds of singularities may occur as a family of Einstein metrics evolves in its moduli space.

**Short bio: **Professor Ruobing Zhang graduated from Princeton University in 2016 and has since held research and teaching positions at Stony Brook University and Princeton University. He has recently moved to University of Wisconsin, Madison for a tenured position.

Professor Zhang is a young leading expert in the study of Einstein metrics, one of the fundamental topics in many areas of geometry. Using various deep technical tools, he has made significant contributions. His works has been published in top journals like Acta Math. and Journal of AMS.

This colloquium is in conjunction with a lunchtime special lecture.

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