Book Review: Elliptic curves

Abstract

REVIEWStheory of fiber bundles and lead naturally to the curvature definition of the characteristic classes.3. Exterior differential systems include any system of partial differential equations.The theory is intrinsic and adapts well to nonlinear equations.This list could continue.Exterior differential calculus is destined to occupy a more important place in multivariable calculus!Of which works was Carian most proud?Through my conversations with him during my Paris séjour of 1936-1937, I would like to venture a guess.I would suggest that it is his works on linear representations, including his discovery of spinors in 1913.He published a book on spinors in 1938 in which he included the physical applications.In the conclusion of the book the authors state: "As a rule, Cartan built his scientific research on works of his predecessors, developing their ideas so well that other mathematicians often forgot the original works."This clearly does not apply to the works mentioned above.Moreover, in the examples used by the authors to illustrate their statement-moving frames and generalized spaces-Cartan developed the methods and ideas for homogeneous spaces with any Lie group, which went far beyond the scope of his predecessors.Cartan roamed through a vast and fertile area of mathematics.With his power and insight he was able to pick up the gems wherever he treaded.His books are full of interesting details.This was not the case with his earlier works, such as pseudogroups and exterior differential systems, which were original but needed clarification.As a result the recognition of his achievements came late.It was perhaps Hermann Weyl's work on group representations in 1925-1926 that made Cartan famous in the general mathematical community.Although Poincaré had a high opinion of the role of a "group" in mathematics and of Cartan's contribution to it (his report on Cartan's work is included in the book), Cartan was elected a member of the French Academy only in 1931.1 I have no doubt that Cartan realized the importance of his works.He was able to ignore the outside reaction and led a simple, happy, and fruitful life.

Locations

• Bulletin of the American Mathematical Society - View - PDF