Ebook: The Mathematics of Frobenius in Context: A Journey Through 18th to 20th Century Mathematics

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.

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Frobenius is best known as creator of the theory of group characters and representations, but his name is attached to a multitude of theorems and concepts from a broad spectrum of mathematical disciplines. In this book his mathematics is presented “in context” in two senses. The first provides the reader with the historical background necessary to understand why Frobenius undertook to solve a particular problem and to appreciate the magnitude of his achievement. Part of the context involves Frobenius’ training in the Berlin school of mathematics presided over by Weierstrass, Kronecker, and Kummer, from whom he learned disciplinary ideals as well as theorems. Frobenius’ mathematics is also presented “in context” in that the author traces the ways in which his work was subsequently applied, developed, and ultimately incorporated into present-day mathematics. As a consequence of the contextual approach, the reader will encounter a broad swath of diverse and important strands of 18th–20th century mathematics, ranging from the work of Lagrange and Laplace on stability of linear systems of differential equations to the theory of complex abelian varieties.

The book is divided into three parts. Part I provides an overview of Frobenius’ entire mathematical career and thus serves as an introduction to the main body of the book. Here, within the framework of his educational and professional career, his contributions to mathematics and the attendant backgrounds are briefly sketched and their subsequent impact upon the development of mathematics indicated. Part II presents the development of core aspects of linear algebra up to and including the work of Weierstrass and Kronecker. The chapters of Part III deal in depth with Frobenius’ major works and can be read independently of one another.

Thomas Hawkins was awarded the MAA Chauvenet Prize for expository writing and was the first recipient of the AMS Whiteman Prize for historical exposition. His last book was Emergence of the Theory of Lie Groups (Springer, 2000).

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Frobenius is best known as creator of the theory of group characters and representations, but his name is attached to a multitude of theorems and concepts from a broad spectrum of mathematical disciplines. In this book his mathematics is presented “in context” in two senses. The first provides the reader with the historical background necessary to understand why Frobenius undertook to solve a particular problem and to appreciate the magnitude of his achievement. Part of the context involves Frobenius’ training in the Berlin school of mathematics presided over by Weierstrass, Kronecker, and Kummer, from whom he learned disciplinary ideals as well as theorems. Frobenius’ mathematics is also presented “in context” in that the author traces the ways in which his work was subsequently applied, developed, and ultimately incorporated into present-day mathematics. As a consequence of the contextual approach, the reader will encounter a broad swath of diverse and important strands of 18th–20th century mathematics, ranging from the work of Lagrange and Laplace on stability of linear systems of differential equations to the theory of complex abelian varieties.

The book is divided into three parts. Part I provides an overview of Frobenius’ entire mathematical career and thus serves as an introduction to the main body of the book. Here, within the framework of his educational and professional career, his contributions to mathematics and the attendant backgrounds are briefly sketched and their subsequent impact upon the development of mathematics indicated. Part II presents the development of core aspects of linear algebra up to and including the work of Weierstrass and Kronecker. The chapters of Part III deal in depth with Frobenius’ major works and can be read independently of one another.

Thomas Hawkins was awarded the MAA Chauvenet Prize for expository writing and was the first recipient of the AMS Whiteman Prize for historical exposition. His last book was Emergence of the Theory of Lie Groups (Springer, 2000).

Content:
Front Matter....Pages i-xiii
Front Matter....Pages 1-1
A Berlin Education....Pages 3-31
Professor at the Zurich Polytechnic: 1874–1892....Pages 33-51
Berlin Professor: 1892–1917....Pages 53-70
Front Matter....Pages 71-71
The Paradigm: Weierstrass’ Memoir of 1858....Pages 73-113
Further Development of the Paradigm: 1858–1874....Pages 115-152
Front Matter....Pages 153-153
The Problem of Pfaff....Pages 155-204
The Cayley–Hermite Problem and Matrix Algebra....Pages 205-246
Arithmetic Investigations: Linear Algebra....Pages 247-281
Arithmetic Investigations: Groups....Pages 283-343
Abelian Functions: Problems of Hermite and Kronecker....Pages 345-385
Frobenius’ Generalized Theory of Theta Functions....Pages 387-431
The Group Determinant Problem....Pages 433-460
Group Characters and Representations 1896–1897....Pages 461-493
Alternative Routes to Representation Theory....Pages 495-514
Characters and Representations After 1897....Pages 515-565
Loose Ends....Pages 567-606
Nonnegative Matrices....Pages 607-649
The Mathematics of Frobenius in Retrospect....Pages 651-657
Back Matter....Pages 659-699

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Frobenius is best known as creator of the theory of group characters and representations, but his name is attached to a multitude of theorems and concepts from a broad spectrum of mathematical disciplines. In this book his mathematics is presented “in context” in two senses. The first provides the reader with the historical background necessary to understand why Frobenius undertook to solve a particular problem and to appreciate the magnitude of his achievement. Part of the context involves Frobenius’ training in the Berlin school of mathematics presided over by Weierstrass, Kronecker, and Kummer, from whom he learned disciplinary ideals as well as theorems. Frobenius’ mathematics is also presented “in context” in that the author traces the ways in which his work was subsequently applied, developed, and ultimately incorporated into present-day mathematics. As a consequence of the contextual approach, the reader will encounter a broad swath of diverse and important strands of 18th–20th century mathematics, ranging from the work of Lagrange and Laplace on stability of linear systems of differential equations to the theory of complex abelian varieties.

The book is divided into three parts. Part I provides an overview of Frobenius’ entire mathematical career and thus serves as an introduction to the main body of the book. Here, within the framework of his educational and professional career, his contributions to mathematics and the attendant backgrounds are briefly sketched and their subsequent impact upon the development of mathematics indicated. Part II presents the development of core aspects of linear algebra up to and including the work of Weierstrass and Kronecker. The chapters of Part III deal in depth with Frobenius’ major works and can be read independently of one another.

Thomas Hawkins was awarded the MAA Chauvenet Prize for expository writing and was the first recipient of the AMS Whiteman Prize for historical exposition. His last book was Emergence of the Theory of Lie Groups (Springer, 2000).

Content:
Front Matter....Pages i-xiii
Front Matter....Pages 1-1
A Berlin Education....Pages 3-31
Professor at the Zurich Polytechnic: 1874–1892....Pages 33-51
Berlin Professor: 1892–1917....Pages 53-70
Front Matter....Pages 71-71
The Paradigm: Weierstrass’ Memoir of 1858....Pages 73-113
Further Development of the Paradigm: 1858–1874....Pages 115-152
Front Matter....Pages 153-153
The Problem of Pfaff....Pages 155-204
The Cayley–Hermite Problem and Matrix Algebra....Pages 205-246
Arithmetic Investigations: Linear Algebra....Pages 247-281
Arithmetic Investigations: Groups....Pages 283-343
Abelian Functions: Problems of Hermite and Kronecker....Pages 345-385
Frobenius’ Generalized Theory of Theta Functions....Pages 387-431
The Group Determinant Problem....Pages 433-460
Group Characters and Representations 1896–1897....Pages 461-493
Alternative Routes to Representation Theory....Pages 495-514
Characters and Representations After 1897....Pages 515-565
Loose Ends....Pages 567-606
Nonnegative Matrices....Pages 607-649
The Mathematics of Frobenius in Retrospect....Pages 651-657
Back Matter....Pages 659-699
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