Authors: A. Borisenko, E. Cabezas-Rivas, V. Miquel

Summary: These lectures are aimed to present mathematical concepts and results necessary for exploring the proof of the Poincaré and Thurston conjectures provided by G. Perelman. A sketch of the proof of both hypotheses is also presented. The following topics are covered: main theorems from the theory of three-dimensional manifolds; the eight three-dimensional homogeneous models of geometry; fundamental statements from the theory of parabolic equations including the maximum principle; Ricci flows on Riemannian manifolds and their singularities; particular features of the Ricci flow on two-dimensional surfaces; Alexandrov spaces and the Gromov-Hausdorff convergence; a survey of G. Perelman’s articles on the Poincaré problem is provided as well.

Reading audience: The lectures are intended for graduate students specializing in physics and mathematics as well as for post-graduate students and scientists of research institutes.

ISBN 978-617-95455-5-9 (online)

Responsible institutions: B. Verkin Institute for Low Temperature Physics and Engineering of the NAS of Ukraine.

Published: Харкiв

Size in pages: 156