The Poincaré conjecture. Clay research conference, resolution of the Poincaré conjecture, Institut Henri Poincaré, Paris, France, June 8--9, 2010 (Q2929566)

This volume contains seven papers based on the Clay Research Conference held at the Institut Henri Poincaré in Paris, in 2010, to celebrate the resolution of Poincaré conjecture. The articles are of expository nature and cover several subjects around geometry and topology of low dimensional manifolds. They are not bound to the resolution of the conjecture and all together provide a broad view of the geometry and topology of manifolds, since Poincaré's time to the current open questions. The volume starts with M. Aliyah's contribution ``Geometry in 2, 3 and 4 dimensions'', which is a brief introduction to geometric low dimensional manifolds.NEWLINENEWLINENEWLINENEWLINENext comes the paper by \textit{J. W. Morgan} entitled [``100 Years of Topology: Work Stimulated by Poincaré's Approach to Classifying Manifolds'', Clay Math. Proc. 19, 7--29 (2014; Zbl 1304.55001)]. As the title explains, it describes how the work of Poincaré influenced many generations of topologists, and how the study of manifolds has evolved during 100 years. It starts with a detailed description of Poincaré's Analysis Situs and its complements, it discusses several developements duringt the 20th Century, and finishes with modern developments in low dimensional topology, where ideas from physics, geometry and analysis appear.NEWLINENEWLINENEWLINENEWLINEThe third contribution is by \textit{C. T. McMullen}: [``The Evolution of Geometric Structures on 3-Manifolds'', Bull. Am. Math. Soc., New Ser. 48, No. 2, 259--274 (2011; Zbl 1214.57017)]. The best description is given by the author himself: ``This article--based on a lecture at that conference--aims to give a brief and impressionistic introduction to the geometrization conjecture: its historical precedents, the approaches to its resolution, and some of the remaining open questions.''NEWLINENEWLINENEWLINENEWLINEThe paper ``Invariants of Manifolds of the Classification Problem'' by \textit{S. K. Donaldson} [Clay Math. Proc. 19, 47--63 (2014; Zbl 1304.57002)] surveys the important progress in the last 25 years on invariants of geometric structures on 3 and 4 dimensional manifolds. In the early 1980's those invariants appeared to be new since they have a different character from those arising from classical algebraic topology. They reflect the role of ideas from geometry and analysis in questions of manifold topology.NEWLINENEWLINENEWLINENEWLINEThe paper entitled ``Volumes of Hyperbolic 3-Manifolds'' by \textit{D. Gabai, R. Meyerhoff} and \textit{P. Milley} surveys work of the authors, including their proof that Week's manifold is the one of minimal volume. The paper goes through Mom's technology and the role of Perelman's work (after Agol, Dunfield, Storm, and Thurston). A list of open questions is also provided.NEWLINENEWLINENEWLINENEWLINE\textit{M. Gromov}, in the paper ``Manifolds: Where Do We Come From? What Are We? Where Are We Going?'', Clay Math. Proc. 19, 81--144 (2014; Zbl 1304.57006)], gives a panoramic view of manifolds in any dimension, and of a geometric approach to topology of manifolds. It is the longest contribution and covers a diverse and broad list of topics, including mathematical biology.NEWLINENEWLINENEWLINENEWLINEFinally \textit{G. Tian} discusses geometric flows on four dimensional manifolds in the paper ``Geometric Analysis on 4-Manifolds'', Clay Math. Proc. 19, 145--166 (2014; Zbl 1304.53002)]. After a brief reminder in dimensions 2 and 3, the article is devoted to questions on geometric structures on 4 manifolds and how they are approached by means of geometric analysis, including several geometric flows.