From 1923-38 **Weyl** evolved the concept of continuous groups using matrix representations. With his application of group theory to quantum mechanics he set up the modern subject.

- His parents were Anna Dieck and Ludwig Weyl who was the director of a bank.
- As a boy Hermann had already showed that he had a great talents for mathematics and for science more generally.
- During this period at Göttingen, Weyl made a reputation for himself as an outstanding mathematician who was producing work which was having a major impact on the progress of mathematics.
- ...Weyl's idea of a space also included the famous separation property later introduced and popularly credited to Felix Hausdorff (1914).
- Weyl himself produced two later editions, the third (and final) of these editions appearing in 1955 covering the same topics as the original text but with a more modern treatment.
- As a privatdozent at Göttingen, Weyl had been influenced by Edmund Husserl who held the chair of philosophy there from 1901 to 1916.
- Language for Weyl held a special importance.
- From 1913 to 1930 Weyl held the chair of mathematics at Zürich Technische Hochschule.
- It was an event which had a large influence on Weyl who quickly became fascinated by the mathematical principles lying behind the theory.
- In 1917 Weyl gave another course presenting an innovative approach to relativity through differential geometry.
- The lectures formed the basis of Weyl's second book Raum-Zeit-Materie Ⓣ(Space-time-matter) which first appeared in 1918 with further editions, each showing how his ideas were developing, in 1919, 1920, and 1923.
- These later ideas included a gauge metric (the Weyl metric) which led to a gauge field theory.
- However Einstein, Pauli, Eddington, and others, did not fully accept Weyl's approach.
- Also over this period Weyl also made contributions on the uniform distribution of numbers modulo 1 which are fundamental in analytic number theory.
- In 1921 Schrödinger was appointed to Zürich where he became a colleague, and soon closest friend, of Weyl.
- From 1923-38 Weyl evolved the concept of continuous groups using matrix representations.
- In particular his theory of representations of semisimple groups, developed during 1924-26, was very deep and considered by Weyl himself to be his greatest achievement.
- The ideas behind this theory had already been introduced by Hurwitz and Schur, but it was Weyl with his general character formula which took them forward.
- From 1930 to 1933 Weyl held the chair of mathematics at Göttingen where he was appointed to fill the vacancy which arose on Hilbert's retirement.
- Here Weyl found a very congenial working environment where he was able to guide and influence the younger generation of mathematicians, a task for which he was admirably suited.
- Weyl remained at the Institute for Advanced Study at Princeton until he retired in 1952.
- Weyl certainly undertook work of major importance at Princeton, but his most productive period was without doubt the years he spent at Zürich.
- Wheeler's theory, like Weyl's, lacks the connection with quantum phenomena that is so important for interactions other than gravitation.
- We have seen above how Weyl's great works were first given as lecture courses.
- Many other great books by Weyl appeared during his years at Princeton.
- There is so much that could be said about all these works, but we restrict ourselves to looking at the contents of Symmetry for this perhaps tells us most about the full range of Weyl's interests.
- In 1951 Weyl retired from the Institute for Advanced Study at Princeton.
- We must say a little about another aspect of Weyl's work which we have not really mentioned, namely his work on mathematical philosophy and the foundations of mathematics.
- It is interesting to note what a large number of the references we quote deal with this aspect of his work and its importance is not only in the work itself but also in the extent to which Weyl's ideas on these topics underlies the rest of his mathematical and physical contributions.
- Weyl was much influenced by Husserl in his outlook and also shared many ideas with Brouwer.

Born 9 November 1885, Elmshorn (near Hamburg), Schleswig-Holstein, Germany. Died 8 December 1955, Zürich, Switzerland.

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Analysis, Group Theory, Origin Germany, Physics

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive