Richard Brauer

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For the American museum director, see Richard H. W. Brauer. Richard Brauer_sentence_0

Not to be confused with L. Richard Brauer_sentence_1 E. J. Brouwer. Richard Brauer_sentence_2

Richard Brauer_table_infobox_0

Richard BrauerRichard Brauer_header_cell_0_0_0
BornRichard Brauer_header_cell_0_1_0 (1901-02-10)February 10, 1901

Charlottenburg, German EmpireRichard Brauer_cell_0_1_1

DiedRichard Brauer_header_cell_0_2_0 April 17, 1977(1977-04-17) (aged 76)

Belmont, Massachusetts, U.S.Richard Brauer_cell_0_2_1

NationalityRichard Brauer_header_cell_0_3_0 German, U.S.Richard Brauer_cell_0_3_1
Alma materRichard Brauer_header_cell_0_4_0 University of Berlin (Ph.D., 1926)Richard Brauer_cell_0_4_1
Known forRichard Brauer_header_cell_0_5_0 Brauer's theorem on induced charactersRichard Brauer_cell_0_5_1
AwardsRichard Brauer_header_cell_0_6_0 Cole Prize in Algebra (1949)

National Medal of Science (1970)Richard Brauer_cell_0_6_1

FieldsRichard Brauer_header_cell_0_7_0 Scientist, mathematicianRichard Brauer_cell_0_7_1
InstitutionsRichard Brauer_header_cell_0_8_0 University of Toronto, University of Michigan, Harvard UniversityRichard Brauer_cell_0_8_1
ThesisRichard Brauer_header_cell_0_9_0 Über die Darstellung der Drehungsgruppe durch Gruppen linearer Substitutionen (1926)Richard Brauer_cell_0_9_1
Doctoral advisorRichard Brauer_header_cell_0_10_0 Issai Schur

Erhard SchmidtRichard Brauer_cell_0_10_1

Doctoral studentsRichard Brauer_header_cell_0_11_0 R. H. Bruck

S. A. Jennings Peter Landrock D. J. Lewis J. Carson Mark Cecil J. Nesbitt Donald S. Passman Ralph Stanton Robert SteinbergRichard Brauer_cell_0_11_1

Richard Dagobert Brauer (February 10, 1901 – April 17, 1977) was a leading German and American mathematician. Richard Brauer_sentence_3

He worked mainly in abstract algebra, but made important contributions to number theory. Richard Brauer_sentence_4

He was the founder of modular representation theory. Richard Brauer_sentence_5

Education and career Richard Brauer_section_0

Alfred Brauer was Richard's brother and seven years older. Richard Brauer_sentence_6

They were born to a Jewish family. Richard Brauer_sentence_7

Both were interested in science and mathematics, but Alfred was injured in combat in World War I. Richard Brauer_sentence_8

As a boy, Richard dreamt of becoming an inventor, and in February 1919 enrolled in Technische Hochschule Berlin-Charlottenburg. Richard Brauer_sentence_9

He soon transferred to University of Berlin. Richard Brauer_sentence_10

Except for the summer of 1920 when he studied at University of Freiburg, he studied in Berlin, being awarded his Ph.D. Richard Brauer_sentence_11

on 16 March 1926. Richard Brauer_sentence_12

Issai Schur conducted a seminar and posed a problem in 1921 that Alfred and Richard worked on together, and published a result. Richard Brauer_sentence_13

The problem also was solved by Heinz Hopf at the same time. Richard Brauer_sentence_14

Richard wrote his thesis under Schur, providing an algebraic approach to irreducible, continuous, finite-dimensional representations of real orthogonal (rotation) groups. Richard Brauer_sentence_15

Ilse Karger also studied mathematics at the University of Berlin; she and Richard were married 17 September 1925. Richard Brauer_sentence_16

Their sons George Ulrich (born 1927) and Fred Gunther (born 1932) also became mathematicians. Richard Brauer_sentence_17

Brauer began his teaching career in Königsberg (now Kaliningrad) working as Konrad Knopp’s assistant. Richard Brauer_sentence_18

Brauer expounded central division algebras over a perfect field while in Königsberg; the isomorphism classes of such algebras form the elements of the Brauer group he introduced. Richard Brauer_sentence_19

When the Nazi Party took over in 1933, the Emergency Committee in Aid of Displaced Foreign Scholars took action to help Brauer and other Jewish scientists. Richard Brauer_sentence_20

Brauer was offered an assistant professorship at University of Kentucky. Richard Brauer_sentence_21

Richard accepted the offer, and by the end of 1933 he was in Lexington, Kentucky, teaching in English. Richard Brauer_sentence_22

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 in the Holocaust. Richard Brauer_sentence_23

Hermann Weyl invited Richard to assist him at Princeton's Institute for Advanced Study in 1934. Richard Brauer_sentence_24

Richard and Nathan Jacobson edited Weyl's lectures Structure and Representation of Continuous Groups. Richard Brauer_sentence_25

Through the influence of Emmy Noether, Richard was invited to University of Toronto to take up a faculty position. Richard Brauer_sentence_26

With his graduate student Cecil J. Nesbitt he developed modular representation theory, published in 1937. Richard Brauer_sentence_27

Robert Steinberg, Stephen Arthur Jennings, and Ralph Stanton were also Brauer’s students in Toronto. Richard Brauer_sentence_28

Brauer also conducted international research with Tadasi Nakayama on representations of algebras. Richard Brauer_sentence_29

In 1941 University of Wisconsin hosted visiting professor Brauer. Richard Brauer_sentence_30

The following year he visited the Institute for Advanced Study and Bloomington, Indiana where Emil Artin was teaching. Richard Brauer_sentence_31

In 1948 Richard and Ilse moved to Ann Arbor, Michigan where he and Robert M. Thrall contributed to the program in modern algebra at University of Michigan. Richard Brauer_sentence_32

With his graduate student K. A. Fowler, Brauer proved the Brauer–Fowler theorem. Richard Brauer_sentence_33

Donald John Lewis was another of his students at University of Michigan. Richard Brauer_sentence_34

In 1952 Brauer joined the faculty of Harvard University. Richard Brauer_sentence_35

Before retiring in 1971 he taught aspiring mathematicians such as Donald Passman and I. Richard Brauer_sentence_36 Martin Isaacs. Richard Brauer_sentence_37

The Brauers frequently traveled to see their friends such as Reinhold Baer, Werner Wolfgang Rogosinski, and Carl Ludwig Siegel. Richard Brauer_sentence_38

Mathematical work Richard Brauer_section_1

Several theorems bear his name, including Brauer's induction theorem, which has applications in number theory as well as finite group theory, and its corollary Brauer's characterization of characters, which is central to the theory of group characters. Richard Brauer_sentence_39

The Brauer–Fowler theorem, published in 1956, later provided significant impetus towards the classification of finite simple groups, for it implied that there could only be finitely many finite simple groups for which the centralizer of an involution (element of order 2) had a specified structure. Richard Brauer_sentence_40

Brauer applied modular representation theory to obtain subtle information about group characters, particularly via his three main theorems. Richard Brauer_sentence_41

These methods were particularly useful in the classification of finite simple groups with low rank Sylow 2-subgroups. Richard Brauer_sentence_42

The Brauer–Suzuki theorem showed that no finite simple group could have a generalized quaternion Sylow 2-subgroup, and the Alperin–Brauer–Gorenstein theorem classified finite groups with wreathed or quasidihedral Sylow 2-subgroups. Richard Brauer_sentence_43

The methods developed by Brauer were also instrumental in contributions by others to the classification program: for example, the Gorenstein–Walter theorem, classifying finite groups with a dihedral Sylow 2-subgroup, and Glauberman's Z* theorem. Richard Brauer_sentence_44

The theory of a block with a cyclic defect group, first worked out by Brauer in the case when the principal block has defect group of order p, and later worked out in full generality by E. Richard Brauer_sentence_45 C. Dade, also had several applications to group theory, for example to finite groups of matrices over the complex numbers in small dimension. Richard Brauer_sentence_46

The Brauer tree is a combinatorial object associated to a block with cyclic defect group which encodes much information about the structure of the block. Richard Brauer_sentence_47

In 1970, he was awarded the National Medal of Science. Richard Brauer_sentence_48

Hypercomplex numbers Richard Brauer_section_2

Main article: Hypercomplex number Richard Brauer_sentence_49

Eduard Study had written an article on hypercomplex numbers for Klein's encyclopedia in 1898. Richard Brauer_sentence_50

This article was expanded for the French language edition by Henri Cartan in 1908. Richard Brauer_sentence_51

By the 1930s there was evident need to update Study’s article, and Richard Brauer was commissioned to write on the topic for the project. Richard Brauer_sentence_52

As it turned out, when Brauer had his manuscript prepared in Toronto in 1936, though it was accepted for publication, politics and war intervened. Richard Brauer_sentence_53

Nevertheless, Brauer kept his manuscript through the 1940s, 1950s, and 1960s, and in 1979 it was published by Okayama University in Japan. Richard Brauer_sentence_54

It also appeared posthumously as paper #22 in the first volume of his Collected Papers. Richard Brauer_sentence_55

His title was "Algebra der hyperkomplexen Zahlensysteme (Algebren)". Richard Brauer_sentence_56

Unlike the articles by Study and Cartan, which were exploratory, Brauer’s article reads as a modern abstract algebra text with its universal coverage. Richard Brauer_sentence_57

Consider his introduction: Richard Brauer_sentence_58

Richard Brauer_description_list_0

  • In the beginning of the 19th century, the usual complex numbers and their introduction through computations with number-pairs or points in the plane, became a general tool of mathematicians. Naturally the question arose whether or not a similar "hypercomplex" number can be defined using points of n-dimensional space. As it turns out, such extension of the system of real numbers requires the concession of some of the usual axioms (Weierstrass 1863). The selection of rules of computation, which cannot be avoided in hypercomplex numbers, naturally allows some choice. Yet in any cases set out, the resulting number systems allow a unique theory with regard to their structural properties and their classification. Further, one desires that these theories stand in close connection with other areas of mathematics, wherewith the possibility of their applications is given.Richard Brauer_item_0_0

While still in Königsberg in 1929, Brauer published an article in Mathematische Zeitschrift "Über Systeme hyperkomplexer Zahlen" which was primarily concerned with integral domains (Nullteilerfrei systeme) and the field theory which he used later in Toronto. Richard Brauer_sentence_59

Publications Richard Brauer_section_3

Richard Brauer_unordered_list_1

  • Brauer, R.; Sah, Chih-han, eds. (1969), , W. A. Benjamin, Inc., New York-Amsterdam, MRRichard Brauer_item_1_1
  • Brauer, R. (1980), Fong, Paul; Wong, Warren J. (eds.), Collected Papers. Vol. I, Mathematicians of Our Time, 17, MIT Press, ISBN 978-0-262-02135-7, MRRichard Brauer_item_1_2
  • Brauer, R. (1980), Fong, Paul; Wong, Warren J. (eds.), , Mathematicians of Our Time, 18, MIT Press, ISBN 978-0-262-02148-7, MRRichard Brauer_item_1_3
  • Brauer, R. (1980), Fong, Paul; Wong, Warren J. (eds.), Collected Papers. Vol. III, Mathematicians of Our Time, 19, MIT Press, ISBN 978-0-262-02149-4, MRRichard Brauer_item_1_4

See also Richard Brauer_section_4

Richard Brauer_unordered_list_2


Credits to the contents of this page go to the authors of the corresponding Wikipedia page: en.wikipedia.org/wiki/Richard Brauer.