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.