Emil Artin was an Austrian mathematician of Armenian descent. Artin was one of the leading mathematicians of the twentieth century, he is best known for his work on algebraic number theory, contributing to class field theory and a new construction of L-functions. He contributed to the pure theories of rings and fields. Emil Artin was born in Vienna to parents Emma Maria, née Laura, a soubrette on the operetta stages of Austria and Germany, Emil Hadochadus Maria Artin, Austrian-born of mixed Austrian and Armenian descent. Several documents, including Emil's birth certificate, list the father's occupation as “opera singer” though others list it as “art dealer.” It seems at least plausible. They were married in St. Stephen's Parish on July 24, 1895. Artin entered school in September 1904 in Vienna. By his father was suffering symptoms of advanced syphilis, among them increasing mental instability, was institutionalized at the established insane asylum at Mauer Öhling, 125 kilometers west of Vienna.
It is notable that neither wife nor child contracted this infectious disease. Artin's father died there July 20, 1906. Young Artin was eight. On July 15, 1907, Artin's mother remarried to a man named Rudolf Hübner: a prosperous manufacturing entrepreneur in the German-speaking city called Reichenberg, Bohemia. Documentary evidence suggests that Emma had been a resident in Reichenberg the previous year, in deference to her new husband, she had abandoned her vocal career. Hübner deemed a life in the theater unseemly unfit for the wife of a man of his position. In September, 1907, Artin entered the Volksschule in Strobnitz, a small town in southern Czechoslovakia near the Austrian border. For that year, he lived away from home; the following year, he returned to the home of his mother and stepfather, entered the Realschule in Reichenberg, where he pursued his secondary education until June, 1916. In Reichenberg, Artin formed a lifelong friendship with a young neighbor, Arthur Baer, who became an astronomer, teaching for many years at Cambridge University.
Astronomy was an interest the two boys shared at this time. They each had telescopes, they rigged a telegraph between their houses, over which once Baer excitedly reported to his friend an astronomical discovery he thought he had made—perhaps a supernova, he thought—and told Artin where in the sky to look. Artin tapped back the terse reply “A-N-D-R-O-M-E-D-A N-E-B-E-L.” Artin's academic performance in the first years at the Realschule was spotty. Up to the end of the 1911–1912 school year, for instance, his grade in mathematics was “genügend,”. Of his mathematical inclinations at this early period he wrote, “Meine eigene Vorliebe zur Mathematik zeigte sich erst im sechzehnten Lebensjahr, während vorher von irgendeiner Anlage dazu überhaupt nicht die Rede sein konnte.” His grade in French for 1912 was “nicht genügend”. He did rather better work in chemistry, but from 1910 to 1912, his grade for “Comportment” was “nicht genügend.” Artin spent the school year 1912–1913 away from home, in France, a period he spoke of as one of the happiest of his life.
He lived that year with the family of Edmond Fritz, in the vicinity of Paris, attended a school there. When he returned from France to Reichenberg, his academic work markedly improved, he began receiving grades of “gut” or “sehr gut” in all subjects—including French and “Comportment.” By the time he completed studies at the Realschule in June, 1916, he was awarded the Reifezeugnis that affirmed him “reif mit Auszeichnung” for graduation to a technical university. Now that it was time to move on to university studies, Artin was no doubt content but to leave Reichenberg, for relations with his stepfather were clouded. According to him, Hübner reproached him “day and night” with being a financial burden, when Artin became a university lecturer and a professor, Hübner deprecated his academic career as self-indulgent and belittled its paltry emolument. In October, 1916, Artin matriculated at the University of Vienna, having focused by now on mathematics, he studied there with Philipp Furtwängler, took courses in astrophysics and Latin.
Studies at Vienna were interrupted when Artin was drafted in 1918 into the Austrian army. Assigned to the K.u. K. 44th Infantry Regiment, he was stationed northwest of Venice at Primolano, on the Italian front in the foothills of the Dolomites. To his great relief, Artin managed to avoid combat by volunteering for service as a translator—his ignorance of Italian notwithstanding, he did know French, of course, some Latin, was a quick study, was motivated by a rational fear in a theater of that war that had proven a meat-grinder. In his scramble to learn at least some Italian, Artin had recourse to an encyclopedia, which he once consulted for help in dealing with the cockroaches that infested the Austrian barracks. At some length, the article described a variety of technical methods, concluding with—Artin laughingly recalled in years—“la caccia diretta". Indeed, “la caccia diretta” was the straightforward method he and his fe
John Torrence Tate Jr. is an American mathematician, distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry and related areas in algebraic geometry. He is professor emeritus at Harvard University, he was awarded the Abel Prize in 2010. Tate was born in Minneapolis, his father, John Tate Sr. was a professor of physics at the University of Minnesota, a longtime editor of Physical Review. His mother, Lois Beatrice Fossler, was a high school English teacher. Tate Jr. received his bachelor's degree in mathematics from Harvard University, entered the doctoral program in physics at Princeton University. He transferred to the mathematics department and received his PhD in 1950 as a student of Emil Artin. Tate taught at Harvard for 36 years before joining the University of Texas in 1990 as a Sid W. Richardson Foundation Regents Chair, he retired from the Texas mathematics department in 2009, returned to Harvard as a professor emeritus. He resides in Cambridge, Massachusetts with his wife Carol.
He has three daughters with his first wife Karin Tate. Tate's thesis on Fourier analysis in number fields has become one of the ingredients for the modern theory of automorphic forms and their L-functions, notably by its use of the adele ring, its self-duality and harmonic analysis on it. Together with his teacher Emil Artin, Tate gave a cohomological treatment of global class field theory, using techniques of group cohomology applied to the idele class group and Galois cohomology; this treatment made more transparent some of the algebraic structures in the previous approaches to class field theory which used central division algebras to compute the Brauer group of a global field. Subsequently, Tate introduced. In the decades following that discovery he extended the reach of Galois cohomology with the Poitou–Tate duality, the Tate–Shafarevich group, relations with algebraic K-theory. With Jonathan Lubin, he recast local class field theory by the use of formal groups, creating the Lubin–Tate local theory of complex multiplication.
He has made a number of individual and important contributions to p-adic theory. He found a p-adic analogue of Hodge theory, now called Hodge–Tate theory, which has blossomed into another central technique of modern algebraic number theory. Other innovations of his include the'Tate curve' parametrization for certain p-adic elliptic curves and the p-divisible groups. Many of his results were not published and some of them were written up by Serge Lang, Jean-Pierre Serre, Joseph H. Silverman and others. Tate and Serre collaborated on a paper on good reduction of abelian varieties; the classification of abelian varieties over finite fields was carried out by Taira Tate. The Tate conjectures are the equivalent for étale cohomology of the Hodge conjecture, they relate to the Galois action on the l-adic cohomology of an algebraic variety, identifying a space of'Tate cycles' that conjecturally picks out the algebraic cycles. A special case of the conjectures, which are open in the general case, was involved in the proof of the Mordell conjecture by Gerd Faltings.
Tate has had a major influence on the development of number theory through his role as a Ph. D. advisor. His students include George Bergman, Bernard Dwork, Benedict Gross, Robert Kottwitz, Jonathan Lubin, Stephen Lichtenbaum, James Milne, V. Kumar Murty, Carl Pomerance, Ken Ribet, Joseph H. Silverman, Dinesh Thakur. In 1956 Tate was awarded the American Mathematical Society's Cole Prize for outstanding contributions to number theory. In 1995 he received the Leroy P. Steele Prize for Lifetime Achievement from the American Mathematical Society, he was awarded a Wolf Prize in Mathematics in 2002/03 for his creation of fundamental concepts in algebraic number theory. In 2012 he became a fellow of the American Mathematical Society. In 2010, the Norwegian Academy of Science and Letters, of which he is a member, awarded him the Abel Prize, citing "his vast and lasting impact on the theory of numbers". According to a release by the Abel Prize committee "Many of the major lines of research in algebraic number theory and arithmetic geometry are only possible because of the incisive contributions and illuminating insights of John Tate.
He has left a conspicuous imprint on modern mathematics."Tate has been described as "one of the seminal mathematicians for the past half-century" by William Beckner, Chairman of the Department of Mathematics at the University of Texas. Tate, Fourier analysis in number fields and Hecke's zeta functions, Princeton University Ph. D. thesis under Emil Artin. Reprinted in Cassels, J. W. S.. Algebraic number theory, London: Academic Press, pp. 305–347, MR 0215665 Tate, John, "The higher dimensional cohomology groups of class field theory", Ann. of Math. 2, 56: 294–297, doi:10.2307/1969801, MR 0049950 Lang, Serge.
Jean-Pierre Serre is a French mathematician who has made contributions to algebraic topology, algebraic geometry, algebraic number theory. He was awarded the Fields Medal in 1954 and the inaugural Abel Prize in 2003. Born in Bages, Pyrénées-Orientales, France, to pharmacist parents, Serre was educated at the Lycée de Nîmes and from 1945 to 1948 at the École Normale Supérieure in Paris, he 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. Their daughter is the former French diplomat and writer Claudine Monteil; the French mathematician Denis Serre is his nephew. He practices skiing, table tennis, rock climbing. From a young age he was an outstanding figure in the school of Henri Cartan, working on algebraic topology, several complex variables and commutative algebra and algebraic geometry, where he introduced sheaf theory and homological algebra techniques.
Serre's thesis concerned 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, made the point that the award 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, on coherent cohomology, Géometrie Algébrique et Géométrie Analytique. At an early stage in his work Serre had perceived 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 couldn't 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 a finite étale map – are important; this acted as one important source of inspiration for Grothendieck to develop étale topology and the corresponding theory of étale cohomology. These tools, developed in full by Grothendieck and collaborators in Séminaire de géométrie algébrique 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" on Galois cohomology. In his paper FAC, Serre asked whether a finitely generated projective module over a polynomial ring is free; this question led to a great deal of activity in commutative algebra, was 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, is the youngest to be awarded 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, 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, he has received many honorary degrees. In 2012 he became a fellow of the American Mathematical Society. Serre has been awarded the highest honors in France as Grand Cross of the Legion of Honour and Grand Cross of the Legion of Merit. List of things named after Jean-Pierre Serre Nicolas Bourbaki Groupes Algébriques et Corps de Classes, translated into English as Algebraic Groups and Class Fields, Springer-Verlag Corps Locaux, Hermann, as Local Fields, Springer-Verlag Cohomologie Galoisienne Collège de France course 1962–63, as Galois Cohomology, Springer-Verlag Algèbre Locale, Multiplicités Collège de France course 1957–58, as Local Algebra, Springer-Verlag "Lie algebras and Lie groups" Harvard Lectures, Springer-Verlag.
Algèbres de Lie Semi-simples Complexes, as Complex Semisimple Lie Algebras, Springer-Verlag Abelian ℓ-Adic Representations and Elliptic Curves, CRC Press, reissue. Addison-Wesley. 1989. Cours d'arithmétique, PUF, as A Course in Arithmetic, Springer-Verlag Représentations linéaires des groupes finis, Hermann, as Linear Represent
Nicholas Michael Katz is an American mathematician, working in algebraic geometry on p-adic methods and moduli problems, number theory. He is a professor of Mathematics at Princeton University and an editor of the journal Annals of Mathematics. Katz graduated from Johns Hopkins University and from Princeton University, where in 1965 he received his master's degree and in 1966 he received his doctorate under supervision of Bernard Dwork with thesis On the Differential Equations Satisfied by Period Matrices. After that, at Princeton, he was an instructor, an assistant professor in 1968, associate professor in 1971 and professor in 1974. From 2002 to 2005 he was the chairman of faculty there, he was a visiting scholar at the University of Minnesota, the University of Kyoto, Paris VI, the Institute for Advanced Study and the IHES. While in France, he adapted methods of scheme theory and category theory to the theory of modular forms. Subsequently, he has applied geometric methods to various exponential sums.
From 1968 to 1969, he was a NATO Postdoctoral Fellow, from 1975 to 1976 and from 1987–1988 Guggenheim Fellow and from 1971 to 1972 Sloan Fellow. In 1978 he was an invited speaker at the International Congress of Mathematicians in Helsinki and 1970 in Nice. Since 2003 he is a member of the American Academy of Arts and Sciences and since 2004 the National Academy of Sciences. In 2003 he was awarded with Peter Sarnak the Levi L. Conant Prize of the American Mathematical Society for the essay "Zeroes of Zeta Functions and Symmetry" in the Bulletin of the American Mathematical Society. Since 2004 he is an editor of the Annals of Mathematics, he played a significant role as a sounding-board for Andrew Wiles when Wiles was developing in secret his proof of Fermat's last theorem. Mathematician and cryptographer Neal Koblitz was one of Katz's students. Katz studied, with Sarnak among others, the connection of the eigenvalue distribution of large random matrices of classical groups to the distribution of the distances of the zeros of various L and zeta functions in algebraic geometry.
He studied trigonometric sums with algebro-geometric methods. He introduced the Katz–Lang finiteness theorem. Gauss sums, Kloosterman sums, monodromy groups. Annals of Mathematical Studies, Princeton 1988. Exponential sums and differential equations. Annals of Mathematical Studies, Princeton 1990. Manuscript with corrections Rigid Local Systems. Annals of Mathematical Studies, Princeton 1996. Twisted L -functions and Monodromy. Annals of Mathematical Studies, Princeton 2002. Moments and Perversity. A Diophantine Perspective. Annals of Mathematical Studies, Princeton 2005, ISBN 0691123306. Convolution and equidistribution: Sato-Tate theorems for finite-field Mellin transforms. Annals of Mathematical Studies, Princeton 2012. With Barry Mazur: Arithmetic Moduli of elliptic curves. Princeton 1985. With Peter Sarnak: Random Matrices, Frobenius Eigenvalues, Monodromy. AMS Colloquium publications 1998, ISBN 0821810170. With Peter Sarnak: "Zeroes of zeta functions and symmetry". Bulletin of the AMS, Vol. 36, 1999, S.1-26.
Nick Katz at the Mathematics Genealogy Project Nick Katz's web page in Princeton
Claude Chevalley was a French mathematician who made important contributions to number theory, algebraic geometry, class field theory, finite group theory, the theory of algebraic groups. He was a founding member of the Bourbaki group, his father, Abel Chevalley, was a French diplomat who, jointly with his wife Marguerite Chevalley née Sabatier, wrote The Concise Oxford French Dictionary. Chevalley graduated from the École Normale Supérieure in 1929, he spent time at the University of Hamburg, studying under Emil Artin, at the University of Marburg, studying under Helmut Hasse. In Germany, Chevalley discovered Japanese mathematics in the person of Shokichi Iyanaga. Chevalley was awarded a doctorate in 1933 from the University of Paris for a thesis on class field theory; when World War II broke out, Chevalley was at Princeton University. After reporting to the French Embassy, he stayed in the USA, first at Princeton at Columbia University, his American students included Leon Ehrenpreis and Gerhard Hochschild.
During his time in the USA, Chevalley became an American citizen and wrote a substantial part of his lifetime output in English. When Chevalley applied for a chair at the Sorbonne, the difficulties he encountered were the subject of a polemical piece by his friend and fellow Bourbakiste André Weil, titled "Science Française?" and published in the NRF. Chevalley was the "professeur B" of the piece, as confirmed in the endnote to the reprint in Weil's collected works, Oeuvres Scientifiques, tome II. Chevalley did obtain a position in 1957 at the faculty of sciences of the University of Paris, after 1970 at the Université de Paris VII. Chevalley had artistic and political interests, was a minor member of the French non-conformists of the 1930s; the following quote by the co-editor of Chevalley's collected works attests to these interests: "Chevalley was a member of various avant-garde groups, both in politics and in the arts... Mathematics was the most important part of his life, but he did not draw any boundary between his mathematics and the rest of his life."
In his PhD thesis, Chevalley made an important contribution to the technical development of class field theory, removing a use of L-functions and replacing it by an algebraic method. At that time use of group cohomology was implicit, cloaked by the language of central simple algebras. In the introduction to André Weil's Basic Number Theory, Weil attributed the book's adoption of that path to an unpublished manuscript by Chevalley. Around 1950, Chevalley wrote a three-volume treatment of Lie groups. A few years he published the work for which he is best remembered, his investigation into what are now called Chevalley groups. Chevalley groups make up 9 of the 18 families of finite simple groups. Chevalley's accurate discussion of integrality conditions in the Lie algebras of semisimple groups enabled abstracting their theory from the real and complex fields; as a consequence, analogues over finite fields could be defined. This was an essential stage in the evolving classification of finite simple groups.
After Chevalley's work, the distinction between "classical groups" falling into the Dynkin diagram classification, sporadic groups which did not, became sharp enough to be useful. What are called'twisted' groups of the classical families could be fitted into the picture. "Chevalley's theorem" refers to his result on the solubility of equations over a finite field. Another theorem of his concerns the constructible sets in algebraic geometry, i.e. those in the Boolean algebra generated by the Zariski-open and Zariski-closed sets. It states. Logicians call this an elimination of quantifiers. In the 1950s, Chevalley led some Paris seminars of major importance: the Séminaire Cartan–Chevalley of the academic year 1955/6, with Henri Cartan, the Séminaire Chevalley of 1956/7 and 1957/8; these dealt with topics on algebraic groups and the foundations of algebraic geometry, as well as pure abstract algebra. The Cartan–Chevalley seminar was the genesis of scheme theory, but its subsequent development in the hands of Alexander Grothendieck was so rapid and inclusive that its historical tracks can appear well covered.
Grothendieck's work subsumed the more specialised contribution of Serre, Goro Shimura, others such as Erich Kähler and Masayoshi Nagata. 1936. L'Arithmetique dans les Algèbres de Matrices. Hermann, Paris. 1940. "La théorie du corps de classes," Annals of Mathematics 41: 394–418. 1946. Theory of Lie groups. Princeton University Press. 1951. "Théorie des groupes de Lie, tome II, Groupes algébriques", Paris. 1951. Introduction to the theory of algebraic functions of one variable, A. M. S. Math. Surveys VI. 1954. The algebraic theory of spinors, Columbia Univ. Press. 1953-1954. Class field theory, Nagoya Univ. 1955. "Théorie des groupes de Lie, tome III, Théorèmes généraux sur les algèbres de Lie", Paris. 1955, "Sur certains groupes simples," Tôhoku Mathematical Journal 7: 14–66. 1955. The construction and study of certain important algebras, Publ. Math. Soc. Japan. 1956. Fundamental concepts of algebra, Acad. Press. 1956-1958. "Classification des groupes de Lie algébriques", Séminaire Chevalley, Secrétariat Math. 11 rue P. Curie, Paris.
Cartier, Springer-Verlag, 2005. 1958. Fondements de la géométrie algébrique, Secrétariat Math. 11 rue P. Curie, Paris. Idèle Valuative criterion of properness Chevalley group Chevalley scheme Chevalley–Iwahori–Nagata theorem Beck–Chevalley condition Non-Conformist Mo