Sources and Studies in the History of Mathematics and Physical Sciences, 2013
From the time of his dissertation (1870) through his work on the problem of Pfaff in 1876 (Chapte... more 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.
Sources and Studies in the History of Mathematics and Physical Sciences, 2013
Ferdinand Georg Frobenius was born in Berlin on 26 October 1849. He was a descendant of a family ... more Ferdinand Georg Frobenius was born in Berlin on 26 October 1849. He was a descendant of a family stemming from Thuringen, 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.
In 1865 when Sophus Lie (1842-1899) completed his studies at the University of Christiania (now O... more In 1865 when Sophus Lie (1842-1899) completed his studies at the University of Christiania (now Oslo), Norway, he had no idea he was destined to become a mathematician. He had done well, but not brilliantly, in all subjects and was toying with the idea of becoming an observational astronomer. He even gave lectures on the subject in the student union. He had a real talent for explaining the geometry of the heavens. To support himself financially while in this state of career indecision he gave private instruction in mathematics. In this connection, Lie began to read the geometrical works by Poncelet, Chasles, and above all Pl~icker. Inspired by his reading, he did some original mathematical research on the real representation of imaginary quantities in projective geometry, a portion of which was accepted for publication by one of the leading mathematics journals of the timeCrelle's journal in Berlin. On the basis of this experience, he decided to devote himself to geometrical research, to become a mathematician. He was 26 years old. Five years later, during the fall of 1873, Lie made a second fateful decision: to devote himself to the enormous task of creating a theory of continuous transformation groups--a task that meant doing mathematics of a quite different sort from the geometrical work that had occupied him in his first years of mathematical research, 1869-1871- a task that ended by occupying most of his creative mathematical energies for the remainder of his career. The purpose of this article is to explain how Lie was led from the one decision to the other. To accomplish this, I have to immerse you in the mathematical world inhabited by Lie, a world that is quite different from the one to which you are accustomed. The first two sections of this article concern the early geometrical work of Lie (18691871), done in close contact with Felix Klein. It was from this work that the ideas emerged which served to redirect Lie's researches. The third section briefly discusses the years 1872-1873, when the theory of first-order PDEs, 6 THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 2 (~ 1994 Springer-Verlag New York particularly in the form given to it by the work of Jacobi, provided what turned out to be a fertile context for the development of the group-related ideas that had emerged during Lie's geometrical investigations. One of the reasons the history of mathematics is fascinating is that it provides insight into the dynamics by means of which ideas and concepts from diverse mathematical theories combine, in remarkable, often unexpected ways, to give birth to entirely new mathematical theories. In the course of showing you how Lie was led from his deci
In 1888 a series of papers began to appear in Mathematische Annalen entitled “The Composition of ... more In 1888 a series of papers began to appear in Mathematische Annalen entitled “The Composition of Continuous Finite Transformation Groups.”’ The appearance of these papers must have raised some eyebrows because they seemed to constitute a major contribution to mathematics and yet the author, Wilhelm Killing, was a little-known, forty-one-year-old Professor at the Lyceum Hosianum in Braunsberg, East Prussia (now a part of Poland). The Lyceum was a Roman Catholic training center for future clergymen. Despite the unlikely background of the author, the papers did in fact constitute a major contribution to mathematics, a contribution as unexpected as it was extraordinary. I t was in these papers that the entire theory of the structure of semisimple Lie algebras originated. Here we find the origins of such key concepts as the rank of an algebra, Cartan subalgebra, Cartan integers, root systems, nilpotent and semisimple algebras, and the radical of an algebra, as well as fundamental results such as the theorem enumerating all possible structures for finite-dimensional simple Lie algebras over the complex field and a radical splitting theorem. The purpose of my talk is to discuss how such an unlikely figure as Killing came to create such unexpected mathematics. As the title suggests, two factors principally determined the direction of Killing’s research. The discoveries in non-Euclidean geometry and the concomitant speculations on the foundations of geometry formed the context of Killing’s work. His contributions to the theory of Lie algebras were a by-product of his research program on the foundations of geometry. But when compared with contemporaneous work on non-Euclidean geometry and its foundations, Killing’s work stands out as atypical. Since it is precisely the peculiar emphasis of Killing’s research program that brings with it the algebraic problem of determining, in effect, all possible structures for Lie algebras, it is of considerable historical interest to seek to understand its basis. In this connection, Killing’s mathematical education is paramount. He
Sources and Studies in the History of Mathematics and Physical Sciences, 2013
From the time of his dissertation (1870) through his work on the problem of Pfaff in 1876 (Chapte... more 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.
Sources and Studies in the History of Mathematics and Physical Sciences, 2013
Ferdinand Georg Frobenius was born in Berlin on 26 October 1849. He was a descendant of a family ... more Ferdinand Georg Frobenius was born in Berlin on 26 October 1849. He was a descendant of a family stemming from Thuringen, 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.
In 1865 when Sophus Lie (1842-1899) completed his studies at the University of Christiania (now O... more In 1865 when Sophus Lie (1842-1899) completed his studies at the University of Christiania (now Oslo), Norway, he had no idea he was destined to become a mathematician. He had done well, but not brilliantly, in all subjects and was toying with the idea of becoming an observational astronomer. He even gave lectures on the subject in the student union. He had a real talent for explaining the geometry of the heavens. To support himself financially while in this state of career indecision he gave private instruction in mathematics. In this connection, Lie began to read the geometrical works by Poncelet, Chasles, and above all Pl~icker. Inspired by his reading, he did some original mathematical research on the real representation of imaginary quantities in projective geometry, a portion of which was accepted for publication by one of the leading mathematics journals of the timeCrelle's journal in Berlin. On the basis of this experience, he decided to devote himself to geometrical research, to become a mathematician. He was 26 years old. Five years later, during the fall of 1873, Lie made a second fateful decision: to devote himself to the enormous task of creating a theory of continuous transformation groups--a task that meant doing mathematics of a quite different sort from the geometrical work that had occupied him in his first years of mathematical research, 1869-1871- a task that ended by occupying most of his creative mathematical energies for the remainder of his career. The purpose of this article is to explain how Lie was led from the one decision to the other. To accomplish this, I have to immerse you in the mathematical world inhabited by Lie, a world that is quite different from the one to which you are accustomed. The first two sections of this article concern the early geometrical work of Lie (18691871), done in close contact with Felix Klein. It was from this work that the ideas emerged which served to redirect Lie's researches. The third section briefly discusses the years 1872-1873, when the theory of first-order PDEs, 6 THE MATHEMATICAL INTELLIGENCER VOL. 16, NO. 2 (~ 1994 Springer-Verlag New York particularly in the form given to it by the work of Jacobi, provided what turned out to be a fertile context for the development of the group-related ideas that had emerged during Lie's geometrical investigations. One of the reasons the history of mathematics is fascinating is that it provides insight into the dynamics by means of which ideas and concepts from diverse mathematical theories combine, in remarkable, often unexpected ways, to give birth to entirely new mathematical theories. In the course of showing you how Lie was led from his deci
In 1888 a series of papers began to appear in Mathematische Annalen entitled “The Composition of ... more In 1888 a series of papers began to appear in Mathematische Annalen entitled “The Composition of Continuous Finite Transformation Groups.”’ The appearance of these papers must have raised some eyebrows because they seemed to constitute a major contribution to mathematics and yet the author, Wilhelm Killing, was a little-known, forty-one-year-old Professor at the Lyceum Hosianum in Braunsberg, East Prussia (now a part of Poland). The Lyceum was a Roman Catholic training center for future clergymen. Despite the unlikely background of the author, the papers did in fact constitute a major contribution to mathematics, a contribution as unexpected as it was extraordinary. I t was in these papers that the entire theory of the structure of semisimple Lie algebras originated. Here we find the origins of such key concepts as the rank of an algebra, Cartan subalgebra, Cartan integers, root systems, nilpotent and semisimple algebras, and the radical of an algebra, as well as fundamental results such as the theorem enumerating all possible structures for finite-dimensional simple Lie algebras over the complex field and a radical splitting theorem. The purpose of my talk is to discuss how such an unlikely figure as Killing came to create such unexpected mathematics. As the title suggests, two factors principally determined the direction of Killing’s research. The discoveries in non-Euclidean geometry and the concomitant speculations on the foundations of geometry formed the context of Killing’s work. His contributions to the theory of Lie algebras were a by-product of his research program on the foundations of geometry. But when compared with contemporaneous work on non-Euclidean geometry and its foundations, Killing’s work stands out as atypical. Since it is precisely the peculiar emphasis of Killing’s research program that brings with it the algebraic problem of determining, in effect, all possible structures for Lie algebras, it is of considerable historical interest to seek to understand its basis. In this connection, Killing’s mathematical education is paramount. He
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