Hermann Weyl
Hermann Weyl  

Born 
Hermann Klaus Hugo Weyl 9 November 1885 Elmshorn, German Empire 
Died 
8 December 1955 Zurich, Switzerland 
(aged 70)
Nationality  German 
Fields  Mathematical physics 
Institutions 
Institute for Advanced Study University of Göttingen ETH Zurich 
Alma mater  University of Göttingen 
Doctoral advisor  David Hilbert^{[1]} 
Doctoral students  Alexander Weinstein 
Other notable students  Saunders Mac Lane 
Known for  See list of topics named after Hermann Weyl 
Influences 
Edmund Husserl^{[2]} L. E. J. Brouwer 
Notable awards  Fellow of the Royal Society^{[3]} 
Spouse  Helene Weyl 
Children  Fritz Joachim Weyl 
Signature 
Hermann Klaus Hugo Weyl, ForMemRS^{[3]} (German: ; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski. His research has had major significance for theoretical physics as well as purely mathematical disciplines including number theory. He was one of the most influential mathematicians of the twentieth century, and an important member of the Institute for Advanced Study during its early years.^{[4]}^{[5]}^{[6]}
Weyl published technical and some general works on space, time, matter, philosophy, logic, symmetry and the history of mathematics. He was one of the first to conceive of combining general relativity with the laws of electromagnetism. While no mathematician of his generation aspired to the 'universalism' of Henri Poincaré or Hilbert, Weyl came as close as anyone. Michael Atiyah, in particular, has commented that whenever he examined a mathematical topic, he found that Weyl had preceded him (The Mathematical Intelligencer (1984), vol.6 no.1).
Contents
 Biography 1

Contributions 2
 Distribution of eigenvalues 2.1
 Geometric foundations of manifolds and physics 2.2
 Topological groups, Lie groups and representation theory 2.3
 Harmonic analysis and analytic number theory 2.4
 Foundations of mathematics 2.5
 Weyl fermions 2.6
 Quotes 3
 Topics named after Hermann Weyl 4
 References 5

Further reading 6
 Primary 6.1
 Secondary 6.2
 External links 7
Biography
Weyl was born in Elmshorn, a small town near Hamburg, in Germany, and attended the gymnasium Christianeum in Altona.^{[7]}
From 1904 to 1908 he studied mathematics and physics in both Göttingen and Munich. His doctorate was awarded at the University of Göttingen under the supervision of David Hilbert whom he greatly admired. After taking a teaching post for a few years, he left Göttingen for Zürich to take the chair of mathematics at the ETH Zurich, where he was a colleague of Albert Einstein, who was working out the details of the theory of general relativity. Einstein had a lasting influence on Weyl who became fascinated by mathematical physics. Weyl met Erwin Schrödinger in 1921, who was appointed Professor at the University of Zürich. They were to become close friends over time. Weyl had some sort of childless love affair with Annemarie (Anny) Schrödinger, while Anny helped raise a daughter whom Erwin had with another woman.^{[8]}
Weyl left Zürich in 1930 to become Hilbert's successor at Göttingen, leaving when the Nazis assumed power in 1933, particularly as his wife was Jewish. He had been offered one of the first faculty positions at the new Institute for Advanced Study in Princeton, New Jersey, but had declined because he did not desire to leave his homeland. As the political situation in Germany grew worse, he changed his mind and accepted when offered the position again. He remained there until his retirement in 1951. Together with his wife, he spent his time in Princeton and Zürich, and died in Zürich in 1955.
Contributions
Distribution of eigenvalues
In 1911 Weyl published Über die asymptotische Verteilung der Eigenwerte (On the asymptotic distribution of eigenvalues) in which he proved that the eigenvalues of the Laplacian in the compact domain are distributed according to the socalled Weyl law. In 1912 he suggested a new proof, based on variational principles. Weyl returned to this topic several times, considered elasticity system and formulated the Weyl conjecture. These works started an important domain—asymptotic distribution of eigenvalues—of modern analysis.
Geometric foundations of manifolds and physics
In 1913, Weyl published Die Idee der Riemannschen Fläche (The Concept of a Riemann Surface), which gave a unified treatment of Riemann surfaces. In it Weyl utilized point set topology, in order to make Riemann surface theory more rigorous, a model followed in later work on manifolds. He absorbed L. E. J. Brouwer's early work in topology for this purpose.
Weyl, as a major figure in the Göttingen school, was fully apprised of Einstein's work from its early days. He tracked the development of relativity physics in his Raum, Zeit, Materie (Space, Time, Matter) from 1918, reaching a 4th edition in 1922. In 1918, he introduced the notion of gauge, and gave the first example of what is now known as a gauge theory. Weyl's gauge theory was an unsuccessful attempt to model the electromagnetic field and the gravitational field as geometrical properties of spacetime. The Weyl tensor in Riemannian geometry is of major importance in understanding the nature of conformal geometry. In 1929, Weyl introduced the concept of the vierbein into general relativity.^{[9]}
His overall approach in physics was based on the phenomenological philosophy of Edmund Husserl, specifically Husserl's 1913 Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie. Erstes Buch: Allgemeine Einführung in die reine Phänomenologie (Ideas of a Pure Phenomenology and Phenomenological Philosophy. First Book: General Introduction). Apparently this was Weyl's way of dealing with Einstein's controversial dependence on the phenomenological physics of Ernst Mach.
Husserl had reacted strongly to Gottlob Frege's criticism of his first work on the philosophy of arithmetic and was investigating the sense of mathematical and other structures, which Frege had distinguished from empirical reference. Hence there is good reason for viewing gauge theory as it developed from Weyl's ideas as a formalism of physical measurement and not a theory of anything physical, i.e. as scientific formalism.
Topological groups, Lie groups and representation theory
From 1923 to 1938, Weyl developed the theory of compact groups, in terms of matrix representations. In the compact Lie group case he proved a fundamental character formula.
These results are foundational in understanding the symmetry structure of quantum mechanics, which he put on a grouptheoretic basis. This included spinors. Together with the mathematical formulation of quantum mechanics, in large measure due to John von Neumann, this gave the treatment familiar since about 1930. Noncompact groups and their representations, particularly the Heisenberg group, were also streamlined in that specific context, in his 1927 Weyl quantization, the best extant bridge between classical and quantum physics to date. From this time, and certainly much helped by Weyl's expositions, Lie groups and Lie algebras became a mainstream part both of pure mathematics and theoretical physics.
His book The Classical Groups, a seminal if difficult text, reconsidered invariant theory. It covered symmetric groups, general linear groups, orthogonal groups, and symplectic groups and results on their invariants and representations.
Harmonic analysis and analytic number theory
Weyl also showed how to use exponential sums in diophantine approximation, with his criterion for uniform distribution mod 1, which was a fundamental step in analytic number theory. This work applied to the Riemann zeta function, as well as additive number theory. It was developed by many others.
Foundations of mathematics
In The Continuum Weyl developed the logic of infinite sets. Weyl appealed in this period to the radical constructivism of the German romantic, subjective idealist Fichte.
Shortly after publishing The Continuum Weyl briefly shifted his position wholly to the intuitionism of Brouwer. In The Continuum, the constructible points exist as discrete entities. Weyl wanted a continuum that was not an aggregate of points. He wrote a controversial article proclaiming that, for himself and L. E. J. Brouwer, "We are the revolution." This article was far more influential in propagating intuitionistic views than the original works of Brouwer himself.
Hegel on the philosophy of nature.^{[10]} Any answer to such a question would be unverifiable, unrelated to experience, and therefore senseless.
However, within a few years Weyl decided that Brouwer's intuitionism did put too great restrictions on mathematics, as critics had always said. The "Crisis" article had disturbed Weyl's formalist teacher Hilbert, but later in the 1920s Weyl partially reconciled his position with that of Hilbert.
After about 1928 Weyl had apparently decided that mathematical intuitionism was not compatible with his enthusiasm for the phenomenological philosophy of Husserl, as he had apparently earlier thought. In the last decades of his life Weyl emphasized mathematics as "symbolic construction" and moved to a position closer not only to Hilbert but to that of Ernst Cassirer. Weyl however rarely refers to Cassirer, and wrote only brief articles and passages articulating this position.
By 1949, Weyl was thoroughly disillusioned with the ultimate value of intuitionism, and wrote: "Mathematics with Brouwer gains its highest intuitive clarity. He succeeds in developing the beginnings of analysis in a natural manner, all the time preserving the contact with intuition much more closely than had been done before. It cannot be denied, however, that in advancing to higher and more general theories the inapplicability of the simple laws of classical logic eventually results in an almost unbearable awkwardness. And the mathematician watches with pain the greater part of his towering edifice which he believed to be built of concrete blocks dissolve into mist before his eyes."
Weyl fermions
In 1929, Weyl proposed a fermion for use in a replacement theory for relativity. This fermion would be a massless quasiparticle and carry electric charge. An electron could be split into two Weyl fermions or formed from two Weyl fermions. Neutrinos were once thought to be Weyl fermions, but they are now known to have mass. Weyl fermions are sought after for electronics applications to solve some problems that electrons present. Such quasiparticles were discovered in 2015, in a form of crystal knowns as Weyl semimetals, a type of topological material.^{[11]}^{[12]}^{[13]}
Quotes
 The question for the ultimate foundations and the ultimate meaning of mathematics remains open; we do not know in which direction it will find its final solution nor even whether a final objective answer can be expected at all. "Mathematizing" may well be a creative activity of man, like language or music, of primary originality, whose historical decisions defy complete objective rationalization.
 —Gesammelte Abhandlungen—as quoted in Year book  The American Philosophical Society, 1943, p. 392
 In these days the angel of topology and the devil of abstract algebra fight for the soul of each individual mathematical domain. Weyl (1939b, p. 500)
Topics named after Hermann Weyl
 Majorana–Weyl spinor
 Schur–Weyl duality
 Weyl algebra
 Weyl basis of the gamma matrices
 Weyl chamber
 Weyl character formula
 Weyl's criterion
 Weyl curvature: see Weyl tensor
 Weyl curvature hypothesis
 Weyl dimension formula, a specialization of the character formula
 Weyl equation, a relativistic wave equation
 Weyl fermion
 Weyl gauge
 Weyl gravity
 Weyl group
 Weyl's inequality
 Weyl integral
 Weyl law
 Weyl's lemma on hypoellipticity
 Weyl's lemma on the "very weak" form of the Laplace equation
 Weyl module
 Weyl notation
 Weyl ordering (Weyl transform)
 Weyl's paradox, properly the Grelling–Nelson paradox
 Weyl's postulate
 Weyl quantization
 Weyl scalar
 Weyl spinor
 Weyl sum, a type of exponential sum
 Weyl symmetry: see Weyl transformation
 Weyl tensor
 Weyl's theorem
 Weyl's theorem on complete reducibility
 Weyl's tile argument
 Weyl transform
 Weyl's unitary trick
 Weyl vector of a compact Lie group
 Peter–Weyl theorem
 Weyl–Schouten theorem
 Weyl transformation
References
 ^ Weyl, H. (1944). "
 ^ Notes to Hermann Weyl (Stanford Encyclopedia of Philosophy)
 ^ ^{a} ^{b}
 ^ .
 ^ Hermann Weyl at the Mathematics Genealogy Project
 ^ Works by or about Hermann Weyl in libraries (WorldCat catalog)
 ^ Elsner, Bernd (2008). "Die Abiturarbeit Hermann Weyls". Christianeum 63 (1): 3–15.
 ^ Moore, Walter (1989). Schrödinger: Life and Thought. Cambridge University Press. pp. 175–176.
 ^ 1929. "Elektron und Gravitation I", Zeitschrift Physik, 56, pp 330–352.
 ^ Gurevich, Yuri. "Platonism, Constructivism and Computer Proofs vs Proofs by Hand", Bulletin of the European Association of Theoretical Computer Science, 1995. This paper describes a letter discovered by Gurevich in 1995 that documents the bet. It is said that when the friendly bet ended, the individuals gathered cited Pólya as the victor (with Kurt Gödel not in concurrence).
 ^ Charles Q. Choi (16 July 2015). "Weyl Fermions Found, a Quasiparticle That Acts Like a Massless Electron". IEEE Spectrum (IEEE).
 ^ "After 85year search, massless particle with promise for nextgeneration electronics found". Science Daily. 16 July 2015.
 ^ SuYang Xu, Ilya Belopolski, Nasser Alidoust, Madhab Neupane, Guang Bian, Chenglong Zhang, Raman Sankar, Guoqing Chang, Zhujun Yuan, ChiCheng Lee, ShinMing Huang, Hao Zheng, Jie Ma, Daniel S. Sanchez, BaoKai Wang, Arun Bansil, Fangcheng Chou, Pavel P. Shibayev, Hsin Lin, Shuang Jia, M. Zahid Hasan. "Discovery of a Weyl Fermion semimetal and topological Fermi arcs". Science.
Further reading
Primary
 1911. Über die asymptotische Verteilung der Eigenwerte, Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 110–117 (1911).
 1913. Idee der Riemannflāche, 2d 1955. The Concept of a Riemann Surface. Addison–Wesley.
 1918. Das Kontinuum, trans. 1987 The Continuum : A Critical Examination of the Foundation of Analysis. ISBN 0486679829
 1918. Raum, Zeit, Materie. 5 edns. to 1922 ed. with notes by Jūrgen Ehlers, 1980. trans. 4th edn. Henry Brose, 1922 Space Time Matter, Methuen, rept. 1952 Dover. ISBN 0486602672.
 1923. Mathematische Analyse des Raumproblems.
 1924. Was ist Materie?
 1925. (publ. 1988 ed. K. Chandrasekharan) Riemann's Geometrische Idee.
 1927. Philosophie der Mathematik und Naturwissenschaft, 2d edn. 1949. Philosophy of Mathematics and Natural Science, Princeton 0689702078. With new introduction by Frank Wilczek, Princeton University Press, 2009, ISBN 9780691141206.
 1928. Gruppentheorie und Quantenmechanik. transl. by H. P. Robertson, The Theory of Groups and Quantum Mechanics, 1931, rept. 1950 Dover. ISBN 0486602699
 1929. "Elektron und Gravitation I", Zeitschrift Physik, 56, pp 330–352. – introduction of the vierbein into GR
 1933. The Open World Yale, rept. 1989 Oxbow Press ISBN 0918024706
 1934. Mind and Nature U. of Pennsylvania Press.
 1934. "On generalized Riemann matrices," Ann. Math. 35: 400–415.
 1935. Elementary Theory of Invariants.
 1935. The structure and representation of continuous groups: Lectures at Princeton university during 1933–34.
 Weyl, Hermann (1939), The Classical Groups. Their Invariants and Representations,
 Weyl, Hermann (1939b), "Invariants",
 1940. Algebraic Theory of Numbers rept. 1998 Princeton U. Press. ISBN 0691059179
 1952. Symmetry. Princeton University Press. ISBN 0691023743
 1968. in K. Chandrasekharan ed, Gesammelte Abhandlungen. Vol IV. Springer.
Secondary
 ed. K. Chandrasekharan,Hermann Weyl, 1885–1985, Centenary lectures delivered by C. N. Yang, R. Penrose, A. Borel, at the ETH Zürich SpringerVerlag, Berlin, Heidelberg, New York, London, Paris, Tokyo – 1986, published for the Eidgenössische Technische Hochschule, Zürich.
 Deppert, Wolfgang et al., eds., Exact Sciences and their Philosophical Foundations. Vorträge des Internationalen HermannWeylKongresses, Kiel 1985, Bern; New York; Paris: Peter Lang 1988,
 Ivor GrattanGuinness, 2000. The Search for Mathematical Roots 18701940. Princeton Uni. Press.
 Erhard Scholz; Robert Coleman; Herbert Korte; Hubert Goenner; Skuli Sigurdsson; Norbert Straumann eds. Hermann Weyl's Raum – Zeit – Materie and a General Introduction to his Scientific Work (Oberwolfach Seminars) (ISBN 3764364769) SpringerVerlag New York, New York, N.Y.
 Thomas Hawkins, Emergence of the Theory of Lie Groups, New York: Springer, 2000.
 Kilmister, C. W. (October 1980), "Zeno, Aristotle, Weyl and Shuard: twoandahalf millennia of worries over number", The Mathematical Gazette (The Mathematical Gazette, Vol. 64, No. 429) 64 (429): 149–158,
 In connection with the Weyl–Pólya bet, a copy of the original letter together with some background can be found in: Pólya, G. (1972). "Eine Erinnerung an Hermann Weyl". Mathematische Zeitschrift 126 (3): 296–298.
External links
 National Academy of Sciences biography
 Bell, John L. Hermann Weyl on intuition and the continuum
 Feferman, Solomon. "Significance of Hermann Weyl's das Kontinuum"
 Straub, William O. Hermann Weyl Website
 Works by Hermann Weyl at Project Gutenberg
 Works by or about Hermann Weyl at Internet Archive
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 1885 births
 1955 deaths
 People from Elmshorn
 People from the Province of SchleswigHolstein
 University of Göttingen alumni
 ETH Zurich faculty
 Institute for Advanced Study faculty
 Foreign Members of the Royal Society
 20thcentury mathematicians
 Differential geometers
 German mathematicians
 German philosophers
 Number theorists
 Pantheists
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 German male writers