The rejuvenation of algebra by the systematic use of the postulational method and the ideas and point of view of abstract group theory has been one of the crowning achievements of twentieth century mathematics. We also hope it will give a solid introduction to this fascinating and mackane growing subject, to those students interested in modern algebra for its own sake. It is a unified birkgoff comprehensive introduction to modern algebra. This is when students should come face-to-face with having to understand universality, or else.
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Upon completing his Harvard BA in , he went to Cambridge University in England to study mathematical physics but switched to studying abstract algebra under Philip Hall.
Birkhoff held no Ph. During the s, Birkhoff, along with his Harvard colleagues Marshall Stone and Saunders Mac Lane , substantially advanced American teaching and research in abstract algebra. A number of papers he wrote in the s, culminating in his monograph, Lattice Theory ; the third edition remains in print , turned lattice theory into a major branch of abstract algebra.
His paper, "On the Structure of Abstract Algebras" founded a new branch of mathematics, universal algebra. Further, in , Birkhoff showed that any equivalence between expressions that holds for all possible forms of operator subject to some equational axioms must have a finite proof using certain underlying rules about equality.
However, frequently it remains undecidable whether or not a particular equivalence holds for the reason that the proofs can become very long. During the war, he worked on radar aiming and ballistics, including the bazooka. In the development of weapons, mathematical questions arose, some of which had not yet been addressed by the literature on fluid dynamics. Birkhoff, a friend of John von Neumann , took a close interest in the rise of the electronic computer.
Birkhoff supervised the Ph. Young on the numerical solution of the partial differential equation of Poisson , in which Young proposed the successive over-relaxation SOR method.
Birkhoff then worked with Richard S. Varga , a former student, who was employed at Bettis Atomic Power Laboratory of the Westinghouse Electronic Corporation in Pittsburgh and was helping to design nuclear reactors. Extending the results of Young, the Birkhoff—Varga collaboration led to many publications on positive operators and iterative methods for p-cyclic matrices.
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