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Walter Feit | |
---|---|

Born | |

Died | July 29, 2004 73) | (aged

Nationality | Austrian American |

Alma mater | University of Chicago University of Michigan |

Known for | Feit–Thompson theorem |

Scientific career | |

Fields | Mathematics |

Doctoral advisor | Robert McDowell Thrall |

**Walter Feit** (October 26, 1930 – July 29, 2004) was an American mathematician who worked in finite group theory and representation theory. His contributions provided elementary infrastructure used in algebra, geometry, topology, number theory, and logic. His work helped the development and utilization of sectors like cryptography, chemistry, and physics.

He was born to a Jewish family in Vienna and escaped for England in 1939 via the Kindertransport.^{ [1] } He moved to the United States in 1946 where he became an undergraduate at the University of Chicago. He did his Ph.D. at the University of Michigan, and became a professor at Cornell University in 1952, and at Yale University in 1964.

His most famous result is his joint, with John G. Thompson, proof of the Feit–Thompson theorem that all finite groups of odd order are solvable. At the time it was written, it was probably the most complicated and difficult mathematical proof ever completed.^{[ according to whom? ]} He wrote almost a hundred other papers, mostly on finite group theory, character theory (in particular introducing the concept of a coherent set of characters), and modular representation theory. Another regular theme in his research was the study of linear groups of small degree, that is, finite groups of matrices in low dimensions. It was often the case that, while the conclusions concerned groups of complex matrices, the techniques employed were from modular representation theory.

He also wrote the books: *The representation theory of finite groups*^{ [2] } and *Characters of finite groups*,^{ [3] } which are now standard references on character theory, including treatments of modular representations and modular characters.

Feit was an invited speaker at the International Congress of Mathematicians (ICM) in Nice in 1970. He was awarded the Cole Prize by the American Mathematical Society in 1965, and was elected to the United States National Academy of Sciences and the American Academy of Arts and Sciences. He also served as Vice-President of the International Mathematical Union.

"In October 2003, on the eve of Professor Feit's retirement, colleagues and former students gathered at Yale for a special four-day "Conference on Groups, Representations and Galois Theory" to honor him and his contributions. Nearly 80 researchers from around the world met to exchange ideas in the fields he had helped to create."^{ [4] }

He died in Branford, Connecticut in 2004 and was survived by his wife, Dr. Sidnie Feit, and a son and daughter.^{ [1] }

"A memorial service was held on Sunday October 10, 2004 at the New Haven Lawn Club, 193 Whitney Avenue, New Haven, CT."

- Feit, Walter (1982).
*The representation theory of finite groups*. North Holland. ISBN 0-444-86155-6. - Feit, Walter (1967).
*Characters of finite groups*. New York: Benjamin.

In mathematics, the **classification of the finite simple groups** is a theorem stating that every finite simple group belongs to one of four broad classes described below. These groups can be seen as the basic building blocks of all finite groups, in a way reminiscent of the way the prime numbers are the basic building blocks of the natural numbers. The Jordan–Hölder theorem is a more precise way of stating this fact about finite groups. However, a significant difference from integer factorization is that such "building blocks" do not necessarily determine a unique group, since there might be many non-isomorphic groups with the same composition series or, put in another way, the extension problem does not have a unique solution.

In mathematics, the **Langlands program** is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by Robert Langlands, it seeks to relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles. Widely seen as the single biggest project in modern mathematical research, the Langlands program has been described by Edward Frenkel as “a kind of grand unified theory of mathematics.”

In abstract algebra, a **finite group** is a group, of which the underlying set contains a finite number of elements.

**Richard Dagobert Brauer** was a leading German and American mathematician. He worked mainly in abstract algebra, but made important contributions to number theory. He was the founder of modular representation theory.

In mathematics, more specifically in group theory, the **character** of a group representation is a function on the group that associates to each group element the trace of the corresponding matrix. The character carries the essential information about the representation in a more condensed form. Georg Frobenius initially developed representation theory of finite groups entirely based on the characters, and without any explicit matrix realization of representations themselves. This is possible because a complex representation of a finite group is determined by its character. The situation with representations over a field of positive characteristic, so-called "modular representations", is more delicate, but Richard Brauer developed a powerful theory of characters in this case as well. Many deep theorems on the structure of finite groups use characters of modular representations.

**John Griggs Thompson** is a mathematician at the University of Florida noted for his work in the field of finite groups. He was awarded the Fields Medal in 1970, the Wolf Prize in 1992 and the 2008 Abel Prize.

**Modular representation theory** is a branch of mathematics, and that part of representation theory that studies linear representations of finite groups over a field *K* of positive characteristic *p*, necessarily a prime number. As well as having applications to group theory, modular representations arise naturally in other branches of mathematics, such as algebraic geometry, coding theory, combinatorics and number theory.

**Nicholas Michael Katz** is an American mathematician, working in algebraic geometry, particularly on *p*-adic methods, monodromy and moduli problems, and number theory. He is currently a professor of Mathematics at Princeton University and an editor of the journal *Annals of Mathematics*.

In mathematics, the **Feit–Thompson theorem**, or **odd order theorem**, states that every finite group of odd order is solvable. It was proved by Walter Feit and John Griggs Thompson.

**James Alexander "Sandy" Green** FRS was a mathematician and Professor at the Mathematics Institute at the University of Warwick, who worked in the field of representation theory.

**Representation theory** is a branch of mathematics that studies abstract algebraic structures by *representing* their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and the algebraic operations in terms of matrix addition and matrix multiplication. The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects helps glean properties and sometimes simplify calculations on more abstract theories.

In mathematics, especially in the area of abstract algebra known as module theory, a **principal indecomposable module** has many important relations to the study of a ring's modules, especially its simple modules, projective modules, and indecomposable modules.

In mathematics, in the theory of finite groups, a **Brauer tree** is a tree that encodes the characters of a block with cyclic defect group of a finite group. In fact, the trees encode the group algebra up to Morita equivalence. Such algebras coming from Brauer trees are called Brauer tree algebras.

In mathematical finite group theory, Thompson's original **uniqueness theorem** states that in a minimal simple finite group of odd order there is a unique maximal subgroup containing a given elementary abelian subgroup of rank 3. Bender (1973) gave a shorter proof of the uniqueness theorem.

In mathematical finite group theory, the **Dade isometry** is an isometry from class functions on a subgroup *H* with support on a subset *K* of *H* to class functions on a group *G*. It was introduced by Dade (1964) as a generalization and simplification of an isometry used by Feit & Thompson (1963) in their proof of the odd order theorem, and was used by Peterfalvi (2000) in his revision of the character theory of the odd order theorem.

**Everett Clarence Dade** is a mathematician at University of Illinois at Urbana–Champaign working on finite groups and representation theory, who introduced the Dade isometry and Dade's conjecture.

**Ronald "Ron“ Mark Solomon** is an American mathematician specializing in the theory of finite groups.

In mathematical representation theory, **coherence** is a property of sets of characters that allows one to extend an isometry from the degree-zero subspace of a space of characters to the whole space. The general notion of coherence was developed by Feit, as a generalization of the proof by Frobenius of the existence of a Frobenius kernel of a Frobenius group and of the work of Brauer and Suzuki on exceptional characters. Feit & Thompson developed coherence further in the proof of the Feit–Thompson theorem that all groups of odd order are solvable.

In the area of modern algebra known as group theory, the **Conway group***Co _{2}* is a sporadic simple group of order

**James Edward Humphreys** is an American mathematician, who works on algebraic groups, Lie groups, and Lie algebras and applications of these mathematical structures. He is known as the author of several mathematical texts, especially *Introduction to Lie Algebras and Representation Theory*.

- 1 2 Scott, Leonard; Solomon, Ronald; Thompson, John; Walter, John; Zelmanov, Efim. "Walter Feit (1930–2004)" (PDF).
*Notices of the American Mathematical Society*.**52**(7): 728–735. - ↑ Fong, Paul (1984). "Review:
*The representation theory of finite groups*, by W. Feit" (PDF).*Bulletin of the American Mathematical Society (N.S.)*.**10**(1): 131–135. doi:10.1090/s0273-0979-1984-15215-7. - ↑ Cohn, Paul M. (1968). "Review:
*Characters of finite groups*by Walter Feit".*Canadian Mathematical Bulletin*.**11**(1): 151–152. - ↑ "Walter Feit. In Memoriam".
*users.math.yale.edu*. Retrieved 2019-10-29.

- O'Connor, John J.; Robertson, Edmund F., "Walter Feit",
*MacTutor History of Mathematics archive*, University of St Andrews . - Walter Feit at the Mathematics Genealogy Project
- Yale obituary
- Walter Feit (1930–2004),
*Notices of the American Mathematical Society*; vol. 52, no. 7 (August 2005). - Walter Feit at Find a Grave

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