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48-566: For example, in 1962 the meeting was dominated by talks on K-theory , at that time the breaking news. The early participants included Jean-Pierre Serre , Jacques Tits , Alexander Grothendieck , Hans Grauert , Nicolaas Kuiper , Raoul Bott , John Milnor , Stephen Smale , Armand Borel , Shiing-Shen Chern , Kunihiko Kodaira , Donald Spencer , Michael Atiyah , Isadore Singer , Shreeram Shankar Abhyankar , Michel Kervaire , Marcel Berger , Karl Stein , Reinhold Remmert , René Thom , Serge Lang and Frank Adams . The institutional structure

96-533: A "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

144-499: 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 . His overall approach in physics

192-675: A French mathematician who has made contributions to algebraic topology , algebraic geometry and algebraic number theory . He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the inaugural Abel Prize in 2003. Born in Bages , Pyrénées-Orientales , to pharmacist parents, Serre was educated at the Lycée de Nîmes. Then he studied at the École Normale Supérieure in Paris from 1945 to 1948. He

240-405: A debate between Husserlian phenomenology and Fichtean constructivism. In the 1920s, before the development of quantum mechanics and inspired by the statistical nature of quantum theory, which was becoming increasingly clear at the time, Weyl turned away from the field-theoretical description of matter towards a theory of active (agens) matter, which was achieved by including the spatial environment in

288-517: A few years, Weyl left Göttingen in 1913 for Zürich to take the chair of mathematics at the ETH Zürich , 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

336-806: A finite étale map – are important. This acted as one important source of inspiration for Grothendieck to develop the étale topology and the corresponding theory of étale cohomology . These tools, developed in full by Grothendieck and collaborators in Séminaire de géométrie algébrique (SGA) 4 and SGA 5, provided the tools for the eventual proof of the Weil conjectures by Pierre Deligne . From 1959 onward Serre's interests turned towards group theory , number theory , in particular Galois representations and modular forms . Amongst his most original contributions were: his " Conjecture II " (still open) on Galois cohomology; his use of group actions on trees (with Hyman Bass );

384-466: A finitely generated projective module over a polynomial ring is free . This question led to a great deal of activity in commutative algebra , and was finally answered in the affirmative by Daniel Quillen and Andrei Suslin independently in 1976. This result is now known as the Quillen–Suslin theorem . Serre, at twenty-seven in 1954, was and still is the youngest person ever to have been awarded

432-415: A form of crystals known as Weyl semimetals , a type of topological material. Weyl had been interested in philosophy since his youth, when he read Immanuel Kant 's "Critique of Pure Reason" with space and time as a priori concepts of knowledge (even if he later disliked Kant's too close ties to Euclidean geometry). From 1912 onwards he was strongly influenced by Edmund Husserl and his phenomenology, which

480-601: 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

528-438: 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

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576-411: A mathematical framework within which his vision of a true continuum, not “synthesized” from discrete elements, is realized. Although the underlying logic of smooth infinitesimal analysis is intuitionistic — the law of excluded middle not being generally affirmable — mathematics developed within avoids the “unbearable awkwardness” to which Weyl refers above." In 1929, Weyl proposed an equation, known as

624-496: A need to construct more general and refined cohomology theories to tackle the Weil conjectures. The problem was that the cohomology of a coherent sheaf over a finite field could not capture as much topology as singular cohomology with integer coefficients. Amongst Serre's early candidate theories of 1954–55 was one based on Witt vector coefficients. Around 1958 Serre suggested that isotrivial principal bundles on algebraic varieties – those that become trivial after pullback by

672-741: A specialist on Johann Gottlieb Fichte. Fichte's Wissenschaftslehre and philosophy, according to which "being" results from the interaction of the "absolute ego" with its material neighbourhood (Umgebung), is also of great influence on Weyl and is reflected in Weyl's use of the neighbourhood concept of topology (continuum). and in Weyl's conception of the general theory of relativity, alongside the influences of Edmund Husserl's phenomenology known directly from Weyl's writings. According to Sieroka, Weyl also finds influences from Gottfried Wilhelm Leibniz's theory of matter (the theory of monads, etc.) and German idealism (Fichte's dialectic) in Weyl's philosophical interpretation of

720-546: 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

768-462: A wealthy family. 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

816-559: Is the former French diplomat, historian and writer Claudine Monteil . The French mathematician Denis Serre is his nephew. He practices skiing, table tennis, and rock climbing (in Fontainebleau ). From a very young age he was an outstanding figure in the school of Henri Cartan , working on algebraic topology , several complex variables and then commutative algebra and algebraic geometry , where he introduced sheaf theory and homological algebra techniques. Serre's thesis concerned

864-790: The Fields Medal . He went on to win the Balzan Prize in 1985, the Steele Prize in 1995, the Wolf Prize in Mathematics in 2000, and was the first recipient of the Abel Prize in 2003. He has been awarded other prizes, such as the Gold Medal of the French National Scientific Research Centre (Centre National de la Recherche Scientifique, CNRS). He is a foreign member of several scientific Academies (US, Norway, Sweden, Russia,

912-509: The Leray–Serre spectral sequence associated to a fibration . Together with Cartan, Serre established the technique of using Eilenberg–MacLane spaces for computing homotopy groups of spheres , which at that time was one of the major problems in topology. In his speech at the Fields Medal award ceremony in 1954, Hermann Weyl gave high praise to Serre, and also made the point that the award

960-530: The Princeton Cemetery . The remains of Hermann's son Michael Weyl (1917–2011) are interred right next to Hermann's ashes in the same columbarium vault. Weyl was a pantheist . 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

1008-588: The University of Zürich . They were to become close friends over time. Weyl had some sort of childless love affair with Schrödinger's wife Annemarie (Anny) Schrödinger (née Bertel), 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

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1056-526: The Weyl equation , for use in a replacement to the Dirac equation . This equation describes massless fermions . A normal Dirac fermion 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. Quasiparticles that behave as Weyl fermions were discovered in 2015, in

1104-517: 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. 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

1152-671: The "last great universal mathematicians of the nineteenth century", Poincaré and Hilbert . Michael Atiyah , in particular, has commented that whenever he examined a mathematical topic, he found that Weyl had preceded him. Hermann Weyl was born in Elmshorn , a small town near Hamburg , in Germany , and attended the Gymnasium Christianeum in Altona . His father, Ludwig Weyl, was a banker; whereas his mother, Anna Weyl (née Dieck), came from

1200-576: 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

1248-550: The Borel–Serre compactification; results on the number of points of curves over finite fields; Galois representations in ℓ-adic cohomology and the proof that these representations have often a "large" image; the concept of p-adic modular form ; and the Serre conjecture (now a theorem) on mod- p representations that made Fermat's Last Theorem a connected part of mainstream arithmetic geometry . In his paper FAC, Serre asked whether

1296-712: The ICM in 1936 at Oslo . He was elected a fellow of the American Physical Society in 1928, a member of the American Academy of Arts and Sciences in 1929, a member of the American Philosophical Society in 1935, and a member of the National Academy of Sciences in 1940. For the academic year 1928–1929, he was a visiting professor at Princeton University , where he wrote a paper, "On a problem in

1344-552: The Legion of Honour (Grand Croix de la Légion d'Honneur) and Grand Cross of the Legion of Merit (Grand Croix de l'Ordre National du Mérite). A list of corrections , and updating, of these books can be found on his home page at Collège de France. Hermann Weyl Hermann Klaus Hugo Weyl , ForMemRS ( German: [vaɪl] ; 9 November 1885 – 8 December 1955) was a German mathematician , theoretical physicist , logician and philosopher . Although much of his working life

1392-598: The Royal Society, Royal Netherlands Academy of Arts and Sciences (1978), American Academy of Arts and Sciences , National Academy of Sciences , the American Philosophical Society ) and has received many honorary degrees (from Cambridge, Oxford, Harvard, Oslo and others). In 2012 he became a fellow of the American Mathematical Society . Serre has been awarded the highest honors in France as Grand Cross of

1440-452: The field theoretical description expressed. He had previously described the general theory of relativity and his own extensions of it, which led to the origin of today's concept of gauge field theories, using differential geometric methods. Under the influence of quantum theory, he turned away from this “geometric field theory”. According to Sieroka, Fichte and Ernst Cassirer were also an important influence in Weyl's late philosophy (science as

1488-451: 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

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1536-400: The physical concept of matter in the context of quantum theory and general relativity and with regard to interaction of a symbol with its surroundings in a mathematical theory structure also in Weyl's philosophy of mathematics (debate between formalism and intuitionism under the influence of Brouwer). He understands the intra-mathematical debate about intuitionism and formalism along the lines of

1584-586: 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. Weyl was cremated in Zürich on December 12, 1955. His ashes remained in private hands until 1999, at which time they were interred in an outdoor columbarium vault in

1632-465: 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, for himself and L. E. J. Brouwer,

1680-403: 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." As John L Bell puts it: "It seems to me a great pity that Weyl did not live to see the emergence in the 1970s of smooth infinitesimal analysis,

1728-413: 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. In 1913, Weyl published Die Idee der Riemannschen Fläche ( The Concept of a Riemann Surface ), which gave

1776-551: The theory of groups arising in the foundations of infinitesimal geometry," 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

1824-503: The twentieth century, and an important member of the Institute for Advanced Study during its early years. Weyl contributed to an exceptionally wide range of fields, including 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 . Freeman Dyson wrote that Weyl alone bore comparison with

1872-854: 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 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

1920-572: 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

1968-576: Was also reflected in some passages in his book “Space, Time, Matter”. In 1927 his contribution Philosophy of Mathematics and Natural Sciences to the Handbook of Philosophy was published by Oldenbourg Verlag, which was later published separately and revised as a book. In an attempt to reconstruct the origins of Hermann Weyl's philosophy and to integrate them into the main currents of philosophy, Norman Sieroka pointed to intensive, long-term discussions between Weyl and his Zurich philosopher colleague Fritz Medicus,

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2016-640: Was awarded his doctorate from the Sorbonne in 1951. From 1948 to 1954 he held positions at the Centre National de la Recherche Scientifique in Paris . In 1956 he was elected professor at the Collège de France , a position he held until his retirement in 1994. His wife, Professor Josiane Heulot-Serre, was a chemist; she also was the director of the Ecole Normale Supérieure de Jeunes Filles. Their daughter

2064-414: 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

2112-584: Was for the first time awarded to a non-analyst. Serre subsequently changed his research focus. In the 1950s and 1960s, a fruitful collaboration between Serre and the two-years-younger Alexander Grothendieck led to important foundational work, much of it motivated by the Weil conjectures . Two major foundational papers by Serre were Faisceaux Algébriques Cohérents (FAC, 1955), on coherent cohomology , and Géométrie Algébrique et Géométrie Analytique ( GAGA , 1956). Even at an early stage in his work Serre had perceived

2160-407: 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 sculptor 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

2208-597: Was reinforced from 1969 by the Sonderforschungsbereich Theoretische Mathematik programme, and from 1980 by the founding of the Max Planck Institute for Mathematics in Bonn. This article about the history of mathematics is a stub . You can help Misplaced Pages by expanding it . Jean-Pierre Serre Jean-Pierre Serre ( French: [sɛʁ] ; born 15 September 1926) is

2256-466: Was spent in Zürich , Switzerland , and then Princeton, New Jersey , he is associated with the University of Göttingen tradition of mathematics, represented by Carl Friedrich Gauss , David Hilbert and Hermann Minkowski . His research has had major significance for theoretical physics as well as purely mathematical disciplines such as number theory . He was one of the most influential mathematicians of

2304-420: 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

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