Talk:Hermann Weyl

I replaced this sentence. It didn't sound right to me:

In 1913, Weyl published Die Idee der Riemannschen Fläche which unified analysis, geometry and topology in the standard model.

-- Walt Pohl 01:40, 2 Mar 2004 (UTC)

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What are specifics of this wager?

George Polya and Weyl, during a mathematicians gathering in Zürich (February 9, 1918), made a bet concerning the least upper bound property. This discussion was on the completely vaguenes of the concepts concerning the construction of the real numbers, sets, and countability [and that the phrases of the least upper bound property is false]. What was in debate was that it may be irrelevant asking about truth of the least upper bound property, other than the basic assertions of Georg Hegel. When the friendly bet ended, the individuals gathered cited Polya as the victor (with Kurt Gödel not in concurrence).

If it's true, it needs to be more specific about the basic assertions of Georg Hegel. Note that neither Polya or Weyl were intuitionists so I doubt very much that any of them doubted the validity of the construction of the real numbers. This sounds very suspicious to me, and unless some reference is provided I think this paragraph should be removed. Moreover, the least upper bound property is not vague; by then it had been well established (for at least 50 years). CSTAR 19:48, 5 Jun 2004 (UTC)

Weyl did look quite closely at intuitionism, trying to connect it with Husserl's work and studying the effect of impredicativity. He gave up, around 1928/9 - this actually seems to me to be quite an important moment in the history of intuitionism. I'm not saying that Weyl was ever a 'true believer', in the Brouwer sense.

Charles Matthews 20:19, 5 Jun 2004 (UTC)

This is certainly interesting and should be icorporated into teh article (and if some specific assertions of Georg Hegel are quoted I'd suggest keeping the assertion on the wager too.)CSTAR 02:06, 6 Jun 2004 (UTC)

Small issues

Does the fragment:

Weyl left the professorship at the Technische Hochschule in Zürich, Switzerland, in the year of 1930 and he became Hilbert's successor at Göttingen where he held the chair of mathematics.

mean

Weyl left the professorship at the Technische Hochschule in Zürich, Switzerland, in the year of 1930 to become Hilbert's successor at Göttingen where he held the chair of mathematics.

I was not sure. By the way, the "in the year of" thing sounds clumsy, and it shows up in many places.

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Does the fragment:

Weyl's research of Riemann surfaces and the associated definition of the complex manifold in one dimension. This is part of the theory of complex manifolds and of differential manifolds.

mean:

Weyl's research was on Riemann surfaces and the associated definition of the complex manifolds in one dimension. This is part of the theory of complex manifolds and of differential manifolds.

I again would have changed myself, but I am not sure.

This article needs some more stylistic edits. I will get to it sometimes. Oleg Alexandrov 21:15, 18 Jan 2005 (UTC)

Weyl's mathematics

I think the article mainly focuses on Weyl's contribution to physics, though he was more important as a mathematician. Unfortunately, I do not have the expertise to edit the article myself.

Weyl's first research, I believe, was in partial differential equations when he was a student of Hilbert. His most important work was in 1925-26 on the representation theory of compact Lie groups, in which he extended the work of Schur, Hurwitz and Cartan. He had a major influence on differential geometry, introducing the global viewpoint on Lie groups and homogenous spaces, which was then pursued vigorously by Elie Cartan. He invented the term "Lie algebra" in a lecture in the 30s, and that lecture inspired Jacobson to work on Lie algebras. Weyl also first defined principal fibre bundles and connections on them.

I am sure he had many more contributions in mathematics, but I know very little. :(

It's not well written, that's for sure. The Lie group work is sort of mentioned. His first research was on operator theory applied to singular cases for things like Sturm-Liouville equations. I'm not sure that's quite right about the topology of Lie groups, connections and bundles; Cartan knew plenty about those, too. Charles Matthews 09:52, 12 Feb 2005 (UTC)

Thanks. Well, let's then wait then for somebody who knows more about those things. Oleg Alexandrov 10:17, 12 Feb 2005 (UTC)

I have done something to make the article read better. (Oh, by the way, since I'm British. I'm not sure that's quite right is a little more assertive than it sounds!) I have just noticed that the introduction is also terrible. Charles Matthews 10:31, 12 Feb 2005 (UTC)

I had missed that fine point about I'm not sure that's quite right. :)

I know some things are not written well here even stylistically. I was not sure about touching the article since I know nothing of the guy. You are welcome to make at least the changes which seem obvious to you. Oleg Alexandrov 10:45, 12 Feb 2005 (UTC)

Hmm... I guess I have to check my sources more carefully then. I thought Weyl was the first person to get any result on the topology of Lie groups. (He proved the finiteness of the fundamental group of a compact Lie group in 1925-26.)

Thanks for editing the article; it looks more comprehensive now. Could you find some way of including the words "Unitary trick" in the section on Lie groups? 210.210.33.46 07:47, 13 Feb 2005 (UTC)

I wouldn't dispute what you say about the fundamental group; it can be read off something in the Weyl group theory, I guess (Stiefel diagram? - it's in Frank Adams' book on Lie groups). The whole business of what Elie Cartan knew and when is interesting and would bear writing up more carefully; and we should have an article topology of Lie groups one day that goes into it. It is probably true that an emerging understanding of Lie groups as global was important - as I recall, Hadamard said he could never hack it. Unitarian trick or such-like: in modern terms it says the unitary group is Zariski-dense in its general linear group? And so the representation theory carries across, in a certain sense? It would be good to support this with some of the other relevant articles on the site. Charles Matthews 08:26, 13 Feb 2005 (UTC)