"Weyl" redirects here. For other persons, see Weyl (surname).
Hermann Klaus Hugo Weyl, ForMemRS (German:[vaɪl]; 9 November 1885 – 8 December 1955) was a Germanmathematician, 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.
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.
Weyl was born in Elmshorn, a small town near Hamburg, in Germany, and attended the gymnasiumChristianeum in Altona.
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.
In September 1913 in Göttingen, Weyl married Friederike Bertha Helene Joseph (March 30, 1893 – September 5, 1948) who went by the name Helene (nickname "Hella"). Helene was a daughter of Dr. Bruno Joseph (December 13, 1861 – June 10, 1934), a physician who held the position of Sanitätsrat in Ribnitz-Damgarten, Germany. Helene was a philosopher (she was a disciple of phenomenologist Edmund Husserl) and also a translator of Spanish literature into German and English (especially the works of Spanish philosopher José Ortega y Gasset). It was through Helene's close connection with Husserl that Hermann became familiar with (and greatly influenced by) Husserl's thought. Hermann and Helene had two sons, Fritz Joachim Weyl (February 19, 1915 – July 20, 1977) and Michael Weyl (September 15, 1917 – March 19, 2011), both of whom were born in Zürich, Switzerland. Helene died in Princeton, New Jersey on September 5, 1948. A memorial service in her honor was held in Princeton on September 9, 1948. Speakers at her memorial service included her son Fritz Joachim Weyl and mathematicians Oswald Veblen and Richard Courant. In 1950 Hermann married sculptress Ellen Bär (née Lohnstein) (April 17, 1902 – July 14, 1988), who was the widow of professor Richard Josef Bär (September 11, 1892 – December 15, 1940) of Zürich.
After taking a teaching post for a few years, Weyl 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. In 1921 Weyl met Erwin Schrödinger, a theoretical physicist who at the time was a professor at the University of Zürich. They were to become close friends over time. Weyl had some sort of childless love affair with Erwin's wife Annemarie (Anny) Schrödinger (née Bertel) (December 31, 1896 – October 3, 1965), while at the same time Anny was helping raise an illegitimate daughter of Erwin's named Ruth Georgie Erica March, who was born in 1934 in Oxford, England.
Weyl was a Plenary Speaker of the International Congress of Mathematicians (ICM) in 1928 at Bologna and an Invited Speaker of the ICM in 1936 at Oslo. For the academic year 1928–1929 he was a visiting professor at Princeton University, where he wrote a paper with Howard P. Robertson.
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 second wife Ellen, he spent his time in Princeton and Zürich, and died from a heart attack on December 8, 1955 while living in Zürich.
Hermann Weyl was cremated in Zurich on December 12, 1955. His cremains remained in private hands until 1999, at which time they were interred in an outdoor columbarium vault in the Princeton Cemetery (aka the Princeton Cemetery of Nassau Presbyterian Church), located at 29 Greenview Avenue, Princeton (Mercer County), New Jersey. The remains of Hermann's son Michael Weyl (1917–2011) are interred right next to Hermann's ashes in the same columbarium vault in the Princeton Cemetery.
Weyl was a pantheist.
Distribution of eigenvalues
Further information: Weyl law, Weyl conjecture
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 so-called 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
Further information: Weyl transformation and Weyl tensor
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.
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).
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.
Topological groups, Lie groups and representation theory
Main articles: Peter–Weyl theorem, Weyl group, Weyl spinor, and Weyl algebra
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 group-theoretic 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. Non-compact 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 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
Further information: Weyl's criterion
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 predicative analysis using the lower levels of Bertrand Russell's ramified theory of types. He was able to develop most of classical calculus, while using neither the axiom of choice nor proof by contradiction, and avoiding Georg Cantor's 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.
George Pólya and Weyl, during a mathematicians' gathering in Zürich (9 February 1918), made a bet concerning the future direction of mathematics. Weyl predicted that in the subsequent 20 years, mathematicians would come to realize the total vagueness of notions such as real numbers, sets, and countability, and moreover, that asking about the truth or falsity of the least upper boundproperty of the real numbers was as meaningful as asking about truth of the basic assertions of Hegel on the philosophy of nature. 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."
Main article: Weyl fermion
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 crystals known as Weyl semimetals, a type of topological material.
- 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)
- 1911. Über die asymptotische Verteilung der Eigenwerte, Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 110–117 (1911).
- 1913. Die Idee der Riemannschen Flāche, 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 0-486-67982-9
- 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 0-486-60267-2.
- 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 978-0-691-14120-6.
- 1928. Gruppentheorie und Quantenmechanik. transl. by H. P. Robertson, The Theory of Groups and Quantum Mechanics, 1931, rept. 1950 Dover. ISBN 0-486-60269-9
- 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 0-918024-70-6
- 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, Princeton University Press, ISBN 978-0-691-05756-9, MR 0000255 
- Weyl, Hermann (1939b), "Invariants", Duke Mathematical Journal, 5 (3): 489–502, doi:10.1215/S0012-7094-39-00540-5, ISSN 0012-7094, MR 0000030
- 1940. Algebraic Theory of Numbers rept. 1998 Princeton U. Press. ISBN 0-691-05917-9
- Weyl, Hermann (1950), "Ramifications, old and new, of the eigenvalue problem", Bull. Amer. Math. Soc., 56: 115–139, doi:10.1090/S0002-9904-1950-09369-0 (text of 1948 Josiah Wilard Gibbs Lecture)
- 1952. Symmetry. Princeton University Press. ISBN 0-691-02374-3
- 1968. in K. Chandrasekharan ed, Gesammelte Abhandlungen. Vol IV. Springer.
Topics named after Hermann Weyl
Main article: List of things named after Hermann Weyl
- ^"Structural Realism": entry by James Ladyman in the Stanford Encyclopedia of Philosophy.
- ^ abNewman, M. H. A. (1957). "Hermann Weyl. 1885-1955". Biographical Memoirs of Fellows of the Royal Society. 3: 305. doi:10.1098/rsbm.1957.0021.
- ^Weyl, H. (1944). "David Hilbert. 1862-1943". Obituary Notices of Fellows of the Royal Society. 4 (13): 547–553. doi:10.1098/rsbm.1944.0006.
- ^Notes to Hermann Weyl (Stanford Encyclopedia of Philosophy).
- ^O'Connor, John J.; Robertson, Edmund F., "Hermann Weyl", MacTutor History of Mathematics archive, University of St Andrews .
- ^Hermann Weyl at the Mathematics Genealogy Project
- ^Works by or about Hermann Weyl in libraries (WorldCat catalog)
- ^Michael Atiyah, The Mathematical Intelligencer (1984), vol. 6 no. 1).
- ^Elsner, Bernd (2008). "Die Abiturarbeit Hermann Weyls". Christianeum. 63 (1): 3–15.
- ^Universität Zũrich Matrikeledition
- ^ Hermann Weyl Collection (AR 3344) (Sys #000195637), Leo Baeck Institute, Center for Jewish History, 15 West 16th Street, New York, NY 10011. The collection includes a typewritten document titled "Hellas letzte Krankheit" ("Hella's Last Illness"); the last sentence on page 2 of the document states: "Hella starb am 5. September , mittags 12 Uhr." ("Hella died at 12:00 Noon on September 5 "). Helene's funeral arrangements were handled by the M. A. Mather Funeral Home (now named the Mather-Hodge Funeral Home), located at 40 Vandeventer Avenue, Princeton, New Jersey. Helene Weyl was cremated on September 6, 1948 at the Ewing Cemetery & Crematory, 78 Scotch Road, Trenton (Mercer County), New Jersey.
- ^For additional information on Helene Weyl, including a bibliography of her translations, published works, and manuscripts, see the following link: "In Memoriam Helene Weyl" by Hermann Weyl. This document, which is one of the items in the Hermann Weyl Collection at the Leo Baeck Institute in New York City, was written by Hermann Weyl at the end of June 1948, about nine weeks before Helene died on September 5, 1948 in Princeton, New Jersey. The first sentence in this document reads as follows: "Eine Skizze, nicht so sehr von Hellas, als von unserem gemeinsamen Leben, niedergeschrieben Ende Juni 1948." ("A sketch, not so much of Hella's life as of our common life, written at the end of June 1948.")
- ^In Memoriam Helene Weyl (1948) by Fritz Joachim Weyl. See: (i) http://www.worldcat.org/oclc/724142550 and (ii) http://d-nb.info/993224164
- ^Ellen Lohnstein and Richard Josef Bär were married on September 14, 1922 in Zürich, Switzerland.
- ^In a photo of Erwin and Annemarie Schrödinger's combined gravemarker in Alpbacher Friedhof (Alpbach Cemetery) located in Alpbach, Kufstein District, Tirol, Austria, the gravemarker gives Annemarie's birthdate as December 31, 1896 — not December 3, 1896 as is mistakenly reported on numerous internet websites — and her deathdate as October 3, 1965. Annemarie's maiden surname was Bertel. Born in Salzburg, Austria, Annemarie was a daughter of Eduard Bertel (1856 – after 1914), a prosperous photographer (court photographer), actor, and industrialist whose various businesses were headquartered in Salzburg. However, it is reported that Eduard moved to Vienna about 1909 or 1910. Annemarie's mother is believed to have been an illegitimate daughter of Georg Junger (1831-1908), a famous and wealthy Salzburg businessman who in 1858 founded a firm of wholesale merchants (selling fashion accessories) located in the Altermarkt section of Salzburg. Annemarie had an older brother named Erich Bertel and a younger sister named Irmgard Bertel.
- ^ Biographical information on Annemarie Schrödinger (née Bertel): Annemarie's birthdate is given as December 3, 1896 on this website, which very likely is wrong since her birthdate as given on her gravemarker in the Alpbach Cemetery is December 31, 1896.
- ^Moore, Walter (1989). Schrödinger: Life and Thought. Cambridge University Press. pp. 175–176. ISBN 0-521-43767-9.
- ^ Ruth Georgie Erica March was born on May 30, 1934 in Oxford, England, but—according to the records presented here—it appears that her birth wasn't "registered" with the British authorities until the 3rd registration quarter (the July–August–September quarter) of the year 1934. Ruth's actual, biological father was Erwin Schrödinger (1887–1961), and her mother was Hildegunde March (née Holzhammer) (born 1900), wife of Austrian physicist Arthur March (February 23, 1891 – April 17, 1957). Hildegunde's friends often called her "Hilde" or "Hilda" rather than Hildegunde. Arthur March was Erwin Schrödinger's assistant at the time of Ruth's birth. The reason Ruth's surname is March (instead of Schrödinger) is because Arthur had agreed to be named as Ruth's father on her birth certificate, even though he wasn't her biological father. Ruth married the engineer Arnulf Braunizer in May 1956, and they have lived in Alpbach, Austria for many years. Ruth has been very active as the sole administrator of the intellectual (and other) property of her father Erwin's estate, which she manages from Alpbach.
- ^"Kontinuierliche Gruppen und ihre Darstellung durch lineare Transformationen von H. Weyl". Atti del Congresso internazionale dei Matematici, Bologna, 1928. Tomo I. Bologna: N. Zanichelli. 1929. pp. 233–246.
- ^Shenstone, Allen G. (24 February 1961). "Princeton & Physics". Princeton Alumni Weekly. 61: pp. 7–8 of article on pp. 6–13 & p. 20.
- ^Robertson, H. P.; Weyl, H. (1929). "On a problem in the theory of groups arising in the foundations of infinitesimal geometry". Bull. Amer. Math. Soc. 35 (5): 686–690. doi:10.1090/S0002-9904-1929-04801-8.
- ^137: Jung, Pauli, and the Pursuit of a Scientific Obsession (New York and London: W. W. Norton & Company, 2009), by Arthur I. Miller (p. 228).
- ^Hermann Weyl's cremains (ashes) are interred in an outdoor columbarium vault in the Princeton Cemetery at this location: Section 3, Block 04, Lot C1, Grave B15.
- ^Hermann Weyl; Peter Pesic. Peter Pesic, ed. Mind and Nature: Selected Writings on Philosophy, Mathematics, and Physics. Princeton University Press. p. 12. ISBN 9780691135458.
- ^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 85-year search, massless particle with promise for next-generation electronics found". Science Daily. 16 July 2015.
- ^Su-Yang Xu; Ilya Belopolski; Nasser Alidoust; Madhab Neupane; Guang Bian; Chenglong Zhang; Raman Sankar; Guoqing Chang; Zhujun Yuan; Chi-Cheng Lee; Shin-Ming 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. 349: 613–617. arXiv:1502.03807. Bibcode:2015Sci...349..613X. doi:10.1126/science.aaa9297.
- ^Moulton, F. R. (1914). "Review: Die Idee der Riemannschen Fläche by Hermann Weyl"(PDF). Bull. Amer. Math. Soc. 20: 384–387. doi:10.1090/s0002-9904-1914-02505-4.
- ^Jacobson, N. (1940). "Review: The Classical Groups by Hermann Weyl"(PDF). Bull. Amer. Math. Soc. 46: 592–595. doi:10.1090/s0002-9904-1940-07236-2.
- ed. K. Chandrasekharan,Hermann Weyl, 1885–1985, Centenary lectures delivered by C. N. Yang, R. Penrose, A. Borel, at the ETH Zürich Springer-Verlag, 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 Hermann-Weyl-Kongresses, Kiel 1985, Bern; New York; Paris: Peter Lang 1988,
- Ivor Grattan-Guinness, 2000. The Search for Mathematical Roots 1870-1940. Princeton Uni. Press.
- Thomas Hawkins, Emergence of the Theory of Lie Groups, New York: Springer, 2000.
- Kilmister, C. W. (October 1980), "Zeno, Aristotle, Weyl and Shuard: two-and-a-half millennia of worries over number", The Mathematical Gazette, The Mathematical Gazette, Vol. 64, No. 429, 64 (429): 149–158, doi:10.2307/3615116, JSTOR 3615116.
- 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. doi:10.1007/BF01110732.
- 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 3-7643-6476-9) Springer-Verlag New York, New York, N.Y.
- Skuli Sigurdsson. "Physics, Life, and Contingency: Born, Schrödinger, and Weyl in Exile." In Mitchell G. Ash, and Alfons Söllner, eds., Forced Migration and Scientific Change: Emigré German-Speaking Scientists and Scholars after 1933 (Washington, D.C.: German Historical Institute and New York: Cambridge University Press, 1996), pp. 48–70.
- Weyl, Hermann (2012), Peter Pesic, ed., Levels of Infinity / Selected Writings on Mathematics and Philosophy, Dover, ISBN 978-0-486-48903-2
Atiyah, M.F., R.H.Bott, and V.K.Patodi. 1973. On the heat equation and the index theorem . Inventiones Math. 19:279–30.
Atiyah, M.F., and R.H.Bott. 1986. A Lefschetz fixed point formula for elliptic and differential operators. Bull. Am. Math. Soc. 72:245– 50.
Bott, R.H. 1988. On induced representations. The mathematical heritage of Herman Weyl. Proc. Symp. Am. Math. Soc. 48:1–14.
Chevalley, C., and A.Weil. 1957. Hermann Weyl (1885–1955) Enseign. Math. III(3).
Donaldson, S.K., and P.B.Kronheimer. 1990. The Geometry of Four-Manifolds. Oxford.
Hodge, W.V.D. 1941. The theory and applications of harmonic integrals. Cambridge.
Mumford, D., J.Fogarty, and F.Kirwan. 1994. Geometric Invariant Theory. Berlin: Springer-Verlag.
Newmann, M.H.A. 1957. Hermann Weyl. In Biographical Memoirs of Fellows of the Royal Society, vol. 3, pp. 305–28.
Pressley, A., and G.B.Segal. 1986. Loop Groups. Oxford.