The Mathematics of Frobenius in Context: A Journey Through 18th to 20th Century Mathematics(Sources and Studies in the History of Mathematics and Physical Sciences)

Frobenius made many important contributions to mathematics in the latter part of the 19th century. Hawkins here focuses on his work in linear algebra and its relationship with the work of Burnside, Cartan, and Molien, and its extension by Schur and Brauer. He also discusses the Berlin school of mathematics and the guiding force of Weierstrass in that school, as well as the fundamental work of d'Alembert, Lagrange, and Laplace, and of Gauss, Eisenstein and Cayley that laid the groundwork for Frobenius's work in linear algebra. The book concludes with a discussion of Frobenius's contribution to the theory of stochastic matrices.

Table of Contents:
1. A Berlin Education.- 2. Professor at the Zurich Polytechnic.- 3. Berlin Professor.- 4. The Paradigm.- 5. Further Development of the Paradigm.- 6. The Problem of Pfaff.- 7. The Cayley-Hermite Problem and Matrix Algebra.- 8. Arithmetical Investigations: Linear Algebra.- 9. Arithmetical Investigations: Groups.- 10. Abelian Functions.- 11. Frobenius' Generalized Theory of Theta Functions.- 12. The Group Determinant Problem.- 13. Group Characters and Representations.- 14. Alternate Routes to Representation Theory.- 15. Characters and Representations after 1897.- 16. Loose Ends.- 17. Nonnegative Matrices.- 18. The Mathematics of Frobenius in Retrospect.- References.- Index.

About the Author :
Thomas Hawkins won the 2001 Whiteman Prize, an AMS prize that honors notable exposition in the history of mathematics. The citation for the prize calls Hawkins "an outstanding historian of mathematics whose current research and numerous publications display the highest standards of mathematical and historical sophistication."  The citation also mentions a number of Hawkins’ works, including his book, The Emergence of the Theory of Lie Groups: An Essay in the History of Mathematics 1869-1926.  "Hawkins’ work has truly transformed our understanding of how modern mathematics has evolved," the citation concludes.

Review :
From the book reviews: “I highly recommend Hawkins’ book. It is very mathematical all the way through. … Hawkins’ work is extraordinarily useful. It allows the mathematical community, even the great majority of us who do not read German well, to understand the work of the very important mathematician Frobenius. The great length of the book is essential to the book’s success.” (David P. Roberts, MAA Reviews, October, 2014) “The author has succeeded admirably in describing the mathematical work of Frobenius. … this book is an excellent contribution to the mathematical literature … it is, or should be, a role model for historical writing, and for bringing the mathematics of the recent past back to life.” (Franz Lemmermeyer, zbMATH, Vol. 1281, 2014)