Geometry of space, physics and analysis
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 develops very powerful a priori estimates and uses the continuity method to solve the complex Monge-Ampère equation, which enables him to prove that there exists a unique Kähler-Einstein metric in each fixed Kähler class. In the fifth section, Yau explains one interesting duality on Calabi-Yau manifolds; physically it is called `T-duality', mathematically known as `Mirror Symmetry'. Roughly speaking, MS is a duality between complex geometry and symplectic geometry on mirror dual CY manifolds. For instance, Kontsevich proposes a homological version of MS as an equivalence of Fukaya category and derived category of coherent sheaves on mirror CY theory. To give a geometric understanding of Kontsevich's work, he proposes how to construct the mirror CY theory using (special) Lagrangian torus fibrations. In the last two sections, he also reviews Yang-Mills theory and singularities in general relativity. To sum up, the article is very well written and it gives many interesting stories on the developments of modern geometry. I think it is an excellent review for young people to have a quick overview of geometry, physics and connections between them.
scientific article; zbMATH DE number 6824443