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Richard Brauer | |
|---|---|
 Richard and Ilse Brauer in 1970 Photo courtesy MFO | |
| Born | (1901-02-10)February 10, 1901 |
| Died | April 17, 1977(1977-04-17) (aged 76) Belmont, Massachusetts, U.S. |
| Alma mater | University of Berlin (PhD, 1926) |
| Known for | Brauer's theorem on induced characters |
| Awards | Cole Prize in Algebra(1949) National Medal of Science(1970) |
| Scientific career | |
| Fields | Scientist,mathematician |
| Institutions | University of Kentucky University of Toronto University of Michigan Harvard University |
| Thesis | Über die Darstellung der Drehungsgruppe durch Gruppen linearer Substitutionen (1926) |
| Doctoral advisor | Issai Schur Erhard Schmidt |
| Doctoral students | R. H. Bruck I. Martin Isaacs S. A. Jennings Peter Landrock D. J. Lewis J. Carson Mark Cecil J. Nesbitt Donald S. Passman Ralph Stanton Robert Steinberg |
Richard Dagobert Brauer (February 10, 1901 – April 17, 1977) was a German and Americanmathematician. He worked mainly inabstract algebra, but made important contributions tonumber theory. He was the founder ofmodular representation theory.
Alfred Brauer was Richard's brother and seven years older. They were born to a Jewish family. Both were interested in science and mathematics, but Alfred was injured in combat in World War I. As a boy, Richard dreamt of becoming an inventor, and in February 1919 enrolled inTechnische Hochschule Berlin-Charlottenburg. He soon transferred toUniversity of Berlin. Except for the summer of 1920 when he studied atUniversity of Freiburg, he studied in Berlin, being awarded hisPhD on 16 March 1926.Issai Schur conducted a seminar and posed a problem in 1921 that Alfred and Richard worked on together, and published a result. The problem also was solved byHeinz Hopf at the same time. Richard wrote his thesis under Schur, providing an algebraic approach toirreducible, continuous,finite-dimensionalrepresentations of realorthogonal (rotation) groups.
Ilse Karger also studied mathematics at the University of Berlin; she and Brauer were married 17 September 1925. Their sons George Ulrich (born 1927) and Fred Gunther (born 1932) also became mathematicians. Brauer began his teaching career inKönigsberg (now Kaliningrad) working asKonrad Knopp’s assistant. Brauer expoundedcentraldivision algebras over aperfect field while in Königsberg; theisomorphism classes of such algebras form the elements of theBrauer group he introduced.
When theNazi Party took over in 1933, theEmergency Committee in Aid of Displaced Foreign Scholars took action to help Brauer and other Jewish scientists.[1] Brauer was offered an assistant professorship atUniversity of Kentucky. Brauer accepted the offer, and by the end of 1933 he was inLexington, Kentucky, teaching in English.[1] Ilse followed the next year with George and Fred; brother Alfred made it to the United States in 1939, but their sister Alice was killed inthe Holocaust.[1]
Hermann Weyl invited Brauer to assist him at Princeton'sInstitute for Advanced Study in 1934. Brauer andNathan Jacobson edited Weyl's lecturesStructure and Representation of Continuous Groups. Through the influence ofEmmy Noether, Brauer was invited toUniversity of Toronto to take up a faculty position. With his graduate studentCecil J. Nesbitt he developedmodular representation theory, published in 1937.Robert Steinberg,Stephen Arthur Jennings, andRalph Stanton were also Brauer’s students in Toronto. Brauer also conducted international research withTadasi Nakayama onrepresentations of algebras. In 1941University of Wisconsin hosted visiting professor Brauer. The following year he visited the Institute for Advanced Study andBloomington, Indiana whereEmil Artin was teaching.
In 1948, Brauer moved toAnn Arbor, Michigan where he andRobert M. Thrall contributed to the program inmodern algebra atUniversity of Michigan.
In 1952, Brauer joined the faculty ofHarvard University and retired in 1971. His students includedDonald John Lewis,Donald Passman, andI. Martin Isaacs. Brauer was elected to theAmerican Academy of Arts and Sciences in 1954,[2] the United StatesNational Academy of Sciences in 1955,[3] and theAmerican Philosophical Society in 1974.[4] The Brauers frequently traveled to see their friends such asReinhold Baer,Werner Wolfgang Rogosinski, andCarl Ludwig Siegel.
Several theorems bear his name, includingBrauer's induction theorem, which has applications innumber theory as well asfinite group theory, and its corollaryBrauer's characterization of characters, which is central to the theory ofgroup characters.
TheBrauer–Fowler theorem, published in 1956, later provided significant impetus towards theclassification of finite simple groups, for it implied that there could only be finitely many finitesimple groups for which thecentralizer of an involution (element oforder 2) had a specified structure.
Brauer introduced the idea of "resolvent degree" in 1975.[5] He appliedmodular representation theory to obtain subtle information about group characters, particularly via histhree main theorems. These methods were particularly useful in the classification of finite simple groups with low rankSylow 2-subgroups. TheBrauer–Suzuki theorem showed that no finite simple group could have ageneralized quaternion Sylow 2-subgroup, and theAlperin–Brauer–Gorenstein theorem classified finite groups with wreathed orquasidihedral Sylow 2-subgroups. The methods developed by Brauer were also instrumental in contributions by others to the classification program: for example, theGorenstein–Walter theorem, classifying finite groups with adihedral Sylow 2-subgroup, andGlauberman'sZ* theorem. The theory of ablock with acyclicdefect group, first worked out by Brauer in the case when theprincipal block has defect group of orderp, and later worked out in full generality byE. C. Dade, also had several applications to group theory, for example to finite groups ofmatrices over thecomplex numbers in small dimension. TheBrauer tree is a combinatorial object associated to a block with cyclic defect group which encodes much information about the structure of the block.
Brauer formulated numerous influential problems[6] onmodular representation theory, among which theBrauer height zero conjecture and thek(B) conjecture.
In 1970, he was awarded theNational Medal of Science.[7]
Eduard Study had written an article on hypercomplex numbers forKlein's encyclopedia in 1898. This article was expanded for theFrench language edition byHenri Cartan in 1908. By the 1930s there was evident need to update Study’s article, and Brauer was commissioned to write on the topic for the project. As it turned out, when Brauer had his manuscript prepared in Toronto in 1936, though it was accepted for publication, politics and war intervened. Nevertheless, Brauer kept his manuscript through the 1940s, 1950s, and 1960s, and in 1979 it was published[8] byOkayama University inJapan. It also appeared posthumously as paper #22 in the first volume of hisCollected Papers. His title was "Algebra der hyperkomplexen Zahlensysteme (Algebren)". Unlike the articles by Study and Cartan, which were exploratory, Brauer’s article reads as a modern abstract algebra text with its universal coverage. Consider his introduction:
While still in Königsberg in 1929, Brauer published an article inMathematische Zeitschrift "Über Systeme hyperkomplexer Zahlen"[9] which was primarily concerned withintegral domains (Nullteilerfrei systeme) and thefield theory which he used later in Toronto.
