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Computers, Reasoning and Mathematical Practice
"... ion in itself is not the goal: for Whitehead [117]"it is the large generalisation, limited by a happy particularity, which is the fruitful conception." As an example consider the theorem in ring theory, which states that if R is a ring, f(x) is a polynomial over R and f(r) = 0 for every element of ..."
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Cited by 6 (2 self)
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ion in itself is not the goal: for Whitehead [117]"it is the large generalisation, limited by a happy particularity, which is the fruitful conception." As an example consider the theorem in ring theory, which states that if R is a ring, f(x) is a polynomial over R and f(r) = 0 for every element of r of R then R is commutative. Special cases of this, for example f(x) is x 2 \Gamma x or x 3 \Gamma x, can be given a first order proof in a few lines of symbol manipulation. The usual proof of the general result [20] (which takes a semester's postgraduate course to develop from scratch) is a corollary of other results: we prove that rings satisfying the condition are semisimple artinian, apply a theorem which shows that all such rings are matrix rings over division rings, and eventually obtain the result by showing that all finite division rings are fields, and hence commutative. This displays von Neumann's architectural qualities: it is "deep" in a way in which the symbol manipulati...
WHITHER MATHEMATICS?
, 2004
"... whither10.tex We describe three successive crises faced by mathematicians during the twentieth century, and their implications for the nature of mathematics. 1 ..."
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whither10.tex We describe three successive crises faced by mathematicians during the twentieth century, and their implications for the nature of mathematics. 1
BETWEEN LOWER AND HIGHER DIMENSIONS (in the work of Terry Lawson)
"... There are several approaches to summarizing a mathematician’s research accomplishments, and each has its advantages and disadvantages. This article is based upon a talk given at Tulane that was aimed at a fairly general audience, including faculty members in other areas and graduate students who had ..."
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There are several approaches to summarizing a mathematician’s research accomplishments, and each has its advantages and disadvantages. This article is based upon a talk given at Tulane that was aimed at a fairly general audience, including faculty members in other areas and graduate students who had taken the usual entry level courses. As such, it is meant to be relatively nontechnical and to emphasize qualitative rather than quantitative issues; in keeping with this aim, references will be given for some standard topological notions that are not normally treated in entry level graduate courses. Since this was an hour talk, it was also not feasible to describe every single piece of published mathematical work that Terry Lawson has ever written; in particular, some papers like [42] and [50] would require lengthy digressions that are not easily related to the central themes in his main lines of research. Instead, we shall focus on some ways in which Terry’s work relates to an important thread in geometric topology; namely, the passage from studying problems in a given dimension to studying problems in the next dimensions. Qualitatively speaking, there are fairly welldeveloped theories for very low dimensions and for all sufficiently large dimensions, but between these ranges there are some dimensions in which the answers to many fundamental
The Classification of the Finite Simple Groups: An Overview
 MONOGRAFÍAS DE LA REAL ACADEMIA DE CIENCIAS DE ZARAGOZA. 26: 89–104, (2004)
, 2004
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DEFORMATIONS OF SIMPLE FINITE GROUP SCHEMES
, 705
"... Abstract. Simple finite group schemes over an algebraically closed field of positive characteristic p ̸ = 2, 3 have been classified. We consider the problem of determining their infinitesimal deformations. In particular, we compute the infinitesimal deformations of the simple finite group schemes of ..."
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Abstract. Simple finite group schemes over an algebraically closed field of positive characteristic p ̸ = 2, 3 have been classified. We consider the problem of determining their infinitesimal deformations. In particular, we compute the infinitesimal deformations of the simple finite group schemes of height one corresponding to the restricted simple Lie algebras. 1.
Simple finite group schemes and their infinitesimal deformations
, 811
"... We show that the classification of simple finite group schemes over an algebraically closed field reduces to the classification of abstract simple finite groups and of simple restricted Lie algebras in positive characteristic. Both these two simple objects have been classified. We review this classi ..."
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We show that the classification of simple finite group schemes over an algebraically closed field reduces to the classification of abstract simple finite groups and of simple restricted Lie algebras in positive characteristic. Both these two simple objects have been classified. We review this classification. Finally, we address the problem of determining the infinitesimal deformations of simple finite group schemes.
In January 1975, Jacques Tits gave a lecture
"... during which he wrote down the order, 2 46 · 3 20 · ..."