Hermann Weyl  

Born  Hermann Klaus Hugo Weyl 9 November 1885 Elmshorn, German Empire 
Died  8 December 1955 Zurich, Switzerland 
(aged 70)
Nationality  German 
Alma mater  University of Göttingen 
Known for  List of topics named after Hermann Weyl Ontic structural realism^{[1]} Wormhole 
Friederike Bertha Helene Joseph (nickname "Hella") (18931948) Ellen Bär (née Lohnstein) (19021988) 

Children  Fritz Joachim Weyl (19151977) Michael Weyl (19172011) 
Awards  Fellow of the Royal Society^{[2]} 
Scientific career  
Fields  Mathematical physics 
Institutions  Institute for Advanced Study University of Göttingen ETH Zurich 
Thesis  Singuläre Integralgleichungen mit besonder Berücksichtigung des Fourierschen Integraltheorems (1908) 
Doctoral advisor  David Hilbert^{[3]} 
Doctoral students  Alexander Weinstein 
Other notable students  Saunders Mac Lane 
Influences  Edmund Husserl^{[4]} L. E. J. Brouwer 
Signature  
Hermann Klaus Hugo Weyl, ForMemRS^{[2]} (German: [va?l]; 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.^{[5]}^{[6]}^{[7]}
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.^{[8]}
Weyl was born in Elmshorn, a small town near Hamburg, in Germany, and attended the gymnasium Christianeum in Altona.^{[9]}
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^{[10]}  September 5, 1948^{[11]}) 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 RibnitzDamgarten, 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).^{[12]} 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),^{[13]} 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 (18801960) and Richard Courant (18881972).^{[14]} In 1950 Hermann married sculptress Ellen Bär (née Lohnstein) (April 17, 1902  July 14, 1988),^{[15]} who was the widow of professor Richard Josef Bär (September 11, 1892  December 15, 1940)^{[16]} 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),^{[17]}^{[18]} 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.^{[19]}^{[20]}
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.^{[21]} His cremains remained in private hands until 1999, at which time they were interred in an outdoor columbarium vault in the Princeton Cemetery^{[22]}^{[23]} (aka the Princeton Cemetery of Nassau Presbyterian Church),^{[24]} located at 29 Greenview Avenue, Princeton (Mercer County), New Jersey. The remains of Hermann's son Michael Weyl (19172011) are interred right next to Hermann's ashes in the same columbarium vault in the Princeton Cemetery.
Weyl was a pantheist.^{[25]}
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 domainasymptotic distribution of eigenvaluesof modern analysis.
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.^{[26]}
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.^{[]}
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 reconsidered invariant theory. It covered symmetric groups, general linear groups, orthogonal groups, and symplectic groups and results on their invariants and representations.
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.
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 bound property of the real numbers was as meaningful as asking about truth of the basic assertions of Hegel on the philosophy of nature.^{[27]} 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."
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.^{[28]}^{[29]}^{[30]}
To use the apt phrase of his son Michael, 'The Open World' (1932) contains "Hermann's dialogues with God" because here the mathematician confronts his ultimate concerns. These do not fall into the traditional religious traditions but are much closer in spirit to Spinoza's rational analysis of what he called "God or nature," so important for Einstein as well. ...In the end, Weyl concludes that this God "cannot and will not be comprehended" by the human mind, even though "mind is freedom within the limitations of existence; it is open toward the infinite." Nevertheless, "neither can God penetrate into man by revelation, nor man penetrate to him by mystical perception."