Grigori Perelman’s landmark proof of the Poincaré Conjecture using Ricci Flow.

: Available through International Press of Boston as part of their "Conference Proceedings and Lecture Notes in Geometry and Topology" series.

This section focuses on the extrinsic geometry of surfaces and higher-dimensional objects embedded in space Differential Calculus of Submanifolds : Foundations of maps and structures Linearization : Introduction to tangent and tensor bundles Curvature and Local Geometry

One of the most celebrated sections of the text deals with scalar curvature. The Yamabe problem asks: Can any smooth, compact Riemannian manifold be deformed conformally to one with constant scalar curvature? The book walks the reader through the analytical techniques (involving critical Sobolev exponents and elliptic PDEs) that ultimately led to the complete resolution of this problem. 3. The Positive Mass Theorem