Geometry of space, physics and analysis
type: Item
instance of: scholarly article
title: Geometry of space, physics and analysis
zbMATH Open document ID: 1384.53064
published in: ICCM Notices
publication date: 2018-01-11T00:00:00Z
review text: The article under review is based on the author's talk delivered at the Tsinghua University in the summer of 2015, on the occasion of the hundredth anniversary of the discovery of general relativity. It gives an overview of many interesting geometries which are related to physics and analysis. In the first section, Yau starts by reviewing the ancient history of geometry in times of Archimedes, Euclidean, etc, and then emphasizing the importance of Newton's work (in 17th century) on differential calculus and its applications to geometry and astronomy. He later mentions: After Newton's groundbreaking work, people like Fermat, Euler, Monge, Gauss, Riemann made fundamental contributions to the subject of differential geometry, where differential calculus is used to study geometric spaces. Notably, Riemann introduced the concept of intrinsic geometry, Riemann surfaces, connectedness of spaces (later known as Betti numbers), etc. The author is also curious whether Riemann himself would have discovered general relativity if he had lived forty years longer. In the second section, he reviews geometry and topology of manifolds, which includes 1) the theory of connections, both for Riemannian manifolds (see works of Levi-Civita, Élie Cartan and Hermann Weyl, etc) and for principal bundles (see works of Yang-Mills, 't Hooft, DeWitt and Faddeev-Popov, etc); 2) the theory of fiber bundles, where Stiefel, Hopf, Ehresmann, Pontryagin, Chern, Steenrod, Leray and Serre made fundamental contributions; 3) applications of calculus to topology (see works of Poincaré, E. Cartan, Weyl, de Rham, Hodge, Kodaira, Morse, Hodge, etc); 4) Erlangen Program proposed by Klein; 5) the theory of Kähler geometry and Hodge theory. In the third and fourth sections, the author reviews his work on solving the Calabi conjecture and the introduction of Calabi-Yau manifolds. CY manifolds are compact Kähler manifolds whose first Chern classes are zero, and they are building blocks in superstring theory. Yau d…
reviewed by: Yalong Cao
Mathematics Subject Classification ID: 53C80
Mathematics Subject Classification ID: 14J32
Mathematics Subject Classification ID: 53-02
zbMATH DE Number: 6824443
zbMATH Keywords: geometry
zbMATH Keywords: physics
zbMATH Keywords: differential calculus
zbMATH Keywords: Calabi-Yau manifolds
author: Shing-Tung Yau
MaRDI profile type: Publication
full work available at URL: iccm.2016.v4.n1.a1
OpenAlex ID: W2503036753
DOI: 10.4310/ICCM.2016.V4.N1.A1
description: scientific article; zbMATH DE number 6824443