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2013 | Buch

A Berlin Education Kapitel lesen Erstes Kapitel lesen

The Mathematics of Frobenius in Context

A Journey Through 18th to 20th Century Mathematics

verfasst von: Thomas Hawkins

Verlag: Springer New York

Buchreihe : Sources and Studies in the History of Mathematics and Physical Sciences

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Overview of Frobenius’ Career and Mathematics

Chapter 1. A Berlin Education
Abstract
Ferdinand Georg Frobenius was born in Berlin on 26 October 1849. He was a descendant of a family stemming from Thüringen, a former state in central Germany and later a part of East Germany. Georg Ludwig Frobenius (1566–1645), a prominent Hamburg publisher of scientific works, including those written by himself on philology, mathematics, and astronomy, was one of his ancestors. His father, Christian Ferdinand, was a Lutheran pastor, and his mother, Christiane Elisabeth Friedrich, was the daughter of a master clothmaker.
Thomas Hawkins
Chapter 2. Professor at the Zurich Polytechnic: 1874–1892
Abstract
By the time the 26-year-old Frobenius arrived at the Zurich Polytechnic Institute, a tradition had already been established whereby a professorship there served as a springboard for promising young German mathematicians en route to a professorship back in Germany. This tradition was unwittingly initiated by Richard Dedekind (1831–1916) in 1858, not long after the polytechnic was founded. As we shall see in Chapters 8 and 9 and in Chapters 12 and 13, Dedekind ranks with Frobenius’ teachers, Weierstrass and Kronecker, as having had a major influence on the directions taken by his mathematics. Some more information about Dedekind is thus in order.
Thomas Hawkins
Chapter 3. Berlin Professor: 1892–1917
Abstract
During Frobenius’ initial years in Berlin (1867–1875), the mathematical leaders, Kummer, Weierstrass, and Kronecker, had worked together in personal and intellectual harmony as illustrated in Chapter 5; but during the 1880s, personal and philosophical differences between Kronecker and Weierstrass emerged. Weierstrass saw Kronecker’s intuitionist views on mathematics as a threat to his own life’s work in analysis, which was based on foundations rejected by Kronecker.
Thomas Hawkins

Berlin-Style Linear Algebra

Frontmatter
Chapter 4. The Paradigm: Weierstrass’ Memoir of 1858
Abstract
In presenting an overview of Frobenius’ career, I indicated that an important component in his education at Berlin involved the work of Kronecker and Weierstrass on the classification of families of quadratic and bilinear forms and the disciplinary ideals their work embodied. Both Weierstrass’ theory of elementary divisors and Kronecker’s generalization of it were inspired by a paper that Weierstrass published in 1858 (Section 4.6). The purpose of this chapter is to sketch the developments that motivated Weierstrass’ work as well as those that provided the means for him to establish it. The former line of development arose from the mathematical analysis of a discrete system of masses oscillating near a point of stable equilibrium. The latter line of development goes back to the mechanics of a rotating rigid body and the existence of principal axes with respect to which the product moments of inertia vanish.
Thomas Hawkins
Chapter 5. Further Development of the Paradigm: 1858–1874
Abstract
In his paper of 1858, Weierstrass had considered pairs of real quadratic forms \(\Phi = {x}^{t}Bx\), \(\Psi = {x}^{t}Ax\), with Φ definite, because his goal was to establish a generalization of the principal axes theorem for them that would provide the basis for a nongeneric treatment of \(B\ddot{y} = Ay\).
Thomas Hawkins

The Mathematics of Frobenius

Frontmatter
Chapter 6. The Problem of Pfaff
Abstract
Having now discussed at length the nature and development of linear algebra at Berlin during Frobenius’ years there, we next turn to Frobenius’ first major paper in which Berlin linear algebra and its concomitant disciplinary ideals played a role. This was also his first paper from Zurich that reflects a break with his work while in Berlin, where it was focused on the theory of ordinary differential equations. The new direction involved what was called the problem of Pfaff. The problem was at the interface of analysis (total differential equations) and algebra and, as perceived by Frobenius, was analogous to the problem of the transformation of quadratic and bilinear forms as treated by Weierstrass and Kronecker. As we shall see, Pfaff’s problem had been around for many years, but work by Clebsch and Natani in the 1860s had revived interest in it. Frobenius’ work was clearly motivated by Clebsch’s treatment of the problem and the issues it suggested vis à vis the disciplinary ideals of the Berlin school. Frobenius’ paper on the problem of Pfaff, which was submitted to Crelle’s Journal in September 1876, marked him as a mathematician of far-ranging ability. His analytic classification theorem (Theorem 6.7) and integrability theorem (Theorem 6.11) have become basic results, and his overall approach by means of the bilinear covariant was to have a great influence on Élie Cartan (Section 6.6), especially as regards his exterior calculus of differential forms and its applications.
Thomas Hawkins
Chapter 7. The Cayley–Hermite Problem and Matrix Algebra
Abstract
Less than 2 years after Frobenius submitted his “monograph” on the problem of Pfaff, he submitted another monograph [181], in which he showed that bilinear forms in n variables (or equivalently, their coefficient matrices) form a linear associative algebra that can be represented symbolically and used to great advantage in solving linear algebraic problems. He was motivated to do so by a problem, called here the Cayley–Hermite problem, which had hitherto been treated on the generic level.
Thomas Hawkins
Chapter 8. Arithmetic Investigations: Linear Algebra
Abstract
From the time of his dissertation (1870) through his work on the problem of Pfaff in 1876 (Chapter 6), Frobenius’ penchant for working on algebraic problems had been pursued within the framework of analysis, especially differential equations. As we saw, his work on the problem of Pfaff—ostensibly a problem within the field of differential equations—had engaged him more fully with linear algebra.
Thomas Hawkins
Chapter 9. Arithmetic Investigations: Groups
Abstract
Frobenius’ interest in arithmetic problems during the late 1870s was not limited to bilinear forms. The groundbreaking work of Gauss on the composition of binary quadratic forms had brought with it a line of thinking that would now be characterized as group-theoretic. This same line of thought resurfaced in Kummer’s revolutionary work on his theory of ideal numbers and prompted Ernst Schering to develop it into what would now be interpreted as the existence part of the fundamental theorem of finite abelian groups, already expressed ambiguously by Schering so as to encompass the finite abelian groups that were implicit in Gauss’ theory of composition of binary quadratic forms as well as those in Kummer’s theory of ideal numbers (ideal class groups). Soon thereafter, both Kronecker and Dedekind expressed Schering’s result explicitly in abstract terms, with Dedekind expressly making the connection with Galois’ notion of a group.
Thomas Hawkins
Chapter 10. Abelian Functions: Problems of Hermite and Kronecker
Abstract
During the 1880s, Frobenius published several papers investigating diverse aspects of the theory of abelian and theta functions. In this and the following chapter, three of these works from 1880 to 1884 will be discussed in detail. (Some other papers from this era dealing with theta functions with half-integer characteristics are considered more tangentially in Chapter 12.)
Thomas Hawkins
Chapter 11. Frobenius’ Generalized Theory of Theta Functions
Abstract
This chapter is devoted to Frobenius’ theory of generalized theta functions, which he called “Jacobian functions” in honor of Jacobi, who had pointed out the fundamental role that can be played by theta functions in establishing the theory of elliptic functions and solving the inversion problem.
Thomas Hawkins
Chapter 12. The Group Determinant Problem
Abstract
This and the following three chapters are devoted to Frobenius’ greatest mathematical achievement, his theory of group characters and representations. The first two chapters consider how he was led to create the basic theory. Then a chapter is devoted to other lines of mathematical thought that led other mathematicians to independently discover at least a part of Frobenius’ results—yet another example of multiple discovery involving Frobenius. The fourth chapter discusses further work by Frobenius on the theory and application of representation theory as well as the contributions made by his best student, I. Schur, and by Schur’s student R. Brauer.
Thomas Hawkins
Chapter 13. Group Characters and Representations 1896–1897
Abstract
Having now established the great appeal to Frobenius of Dedekind’s suggestion that he study Θ and its factorization, let us consider, with the aid of his correspondence with Dedekind, how he progressed. His first progress report to Dedekind came in a letter dated 12 April 1896, just nine days after Dedekind had finished writing his letter to Frobenius.
Thomas Hawkins
Chapter 14. Alternative Routes to Representation Theory
Abstract
The correspondence between Dedekind and Frobenius makes it clear that if Dedekind had not decided to introduce and study group determinants—a subject with no established tradition and really outside his main interests in algebraic number theory—or if he had decided not to communicate his ideas on group determinants to Frobenius, especially given Frobenius’ complete lack of curiosity about Dedekind’s allusion to a connection between hypercomplex numbers and groups, it is unlikely that Frobenius would be known as the creator of the theory of group characters and representations. This is not to say that the theory would have remained undiscovered for a long time. On the contrary, three lines of mathematical investigation were leading to essentially the same theory that Frobenius had begun to explore: (1) the theory of noncommutative hypercomplex number systems; (2) Lie’s theory of continuous groups; and (3) Felix Klein’s research program on a generalized Galois theory. The main purpose of this chapter is to briefly indicate how these lines of investigation were leading—or in some cases did lead—to the results of Frobenius’ theory.
Thomas Hawkins
Chapter 15. Characters and Representations After 1897
Abstract
Frobenius’ papers of 1896–1897 marked the beginning of a new theory, a theory that continued to evolve in various directions for over a half-century. Frobenius himself, along with Burnside, made significant contributions to the theory after 1897, and many new ideas, viewpoints, and directions were introduced by Frobenius’ student Issai Schur (1875–1941), and then by Schur’s student Richard Brauer (1901–1977). In this chapter, these later developments will be sketched, with particular emphasis on matters that relate to the presentation in the previous sections.
Thomas Hawkins
Chapter 16. Loose Ends
Abstract
As its title suggests, this chapter is devoted to tying up several historical loose ends related to the work of Frobenius featured in the previous chapters. The first section focuses on work done by Frobenius in response to the discovery of a gap in Weierstrass’ theory of elementary divisors as it applied to families of quadratic forms. Frobenius gave two solutions to the problem of filling the gap. The first drew upon the results and analogical reasoning used in his arithmetic theory of bilinear forms and its application to elementary divisor theory (Chapter 8).
Thomas Hawkins
Chapter 17. Nonnegative Matrices
Abstract
This final chapter on Frobenius’ mathematics is devoted to the paper he submitted to the Berlin Academy on 23 May 1912 with the title “On matrices with nonnegative elements” [231].
Thomas Hawkins
Chapter 18. The Mathematics of Frobenius in Retrospect
Abstract
In terms of their approach to creative work, mathematicians display a spectrum of tendencies. Some focus most of their time and effort on building up a monumental theory. Sophus Lie was such a mathematician, with his focus on his theory of transformation groups. Among Frobenius’ mentors, Weierstrass, with his focus on the theory of abelian integrals and functions and the requisite foundations in complex function theory, and Richard Dedekind, with his theory of algebraic numbers and ideals, are further examples of mathematicians who were primarily theory builders. At the other end of the spectrum are mathematicians whose focus was first and foremost on concrete mathematical problems. Of course, many mathematicians fall somewhere between these extremes. A prime example is Hilbert, who created several far-reaching theories, such as his theory of integral equations, but also solved many specific problems, such as the finite basis problem in the theory of invariants, Waring’s problem, and Dirichlet’s problem; and of course he posed his famous 23 mathematical problems for others to attempt to solve. Frobenius was decidedly at the problem-solver end of the spectrum. Virtually all of his important mathematical achievements were driven by the desire to solve specific mathematical problems, not famous long-standing problems such as Waring’s problem, but in general, problems that he perceived in the mathematics of his time.
Thomas Hawkins
Backmatter
Metadaten
Titel
The Mathematics of Frobenius in Context
verfasst von
Thomas Hawkins
Copyright-Jahr
2013
Verlag
Springer New York
Electronic ISBN
978-1-4614-6333-7
Print ISBN
978-1-4614-6332-0
DOI
https://doi.org/10.1007/978-1-4614-6333-7

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