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14
Zariski geometries
 Journal of the American Mathematical Society
, 1996
"... Let k be an algebraically closed field. The set of ordered ntuples from k is viewed as an ndimensional space; a subset described by the vanishing of a polynomial, or a family of polynomials, is called an algebraic set,oraZariski closed set. Algebraic geometry describes the behavior of these sets. ..."
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Let k be an algebraically closed field. The set of ordered ntuples from k is viewed as an ndimensional space; a subset described by the vanishing of a polynomial, or a family of polynomials, is called an algebraic set,oraZariski closed set. Algebraic geometry describes the behavior of these sets. The goal of this paper is a converse.
THE AUTOMORPHISM GROUP OF THE COMBINATORIAL GEOMETRY OF AN ALGEBRAICALLY CLOSED FIELD
"... Throughout this paper L will be an algebraically closed field and K an algebraically closed subfield such that the transcendence rank of L over K is at least 3. In [4] we considered the pregeometry G(L/K) = (L,clK), where, for Y a subset of L, the closure clK ( Y) is the algebraic closure in L of t ..."
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Throughout this paper L will be an algebraically closed field and K an algebraically closed subfield such that the transcendence rank of L over K is at least 3. In [4] we considered the pregeometry G(L/K) = (L,clK), where, for Y a subset of L, the closure clK ( Y) is the algebraic closure in L of the subfield generated by K and
Cosets, genericity, and the Weyl group
 J. Algebra
"... In a connected group of finite Morley rank in which, generically, elements belong to connected nilpotent subgroups, proper normalizing cosets of definable subgroups are not generous. We explain why this is true and what consequences this has on an abstract theory of Weyl groups in groups of finite M ..."
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In a connected group of finite Morley rank in which, generically, elements belong to connected nilpotent subgroups, proper normalizing cosets of definable subgroups are not generous. We explain why this is true and what consequences this has on an abstract theory of Weyl groups in groups of finite Morley rank. The only known infinite simple groups of finite Morley rank are the simple algebraic groups over algebraically closed fields and this is a motivation, among many others, for a classification project of these groups. It borrows ideas and techniques from the Classification of the Finite Simple Groups but at the same time it may provide, sometimes, a kind of simplified version of the finite case. This is mostly due to the existence of wellbehaved notions of genericity and connectivity in the infinite case, which unfortunately have no direct finite analogs. The present note deals with a very specific and technical topic concerning such arguments based on genericity in the case of infinite groups of finite Morley
Groups in stable and simple theories
, 2010
"... were written after the lecture, but have not been carefully polished. The bibliography is not carefully put together. For an old but wonderful introduction to stable group theory, see [13]. For an update, often with results stated in great generality, see [15]. On groups of finite Morley rank, [2] g ..."
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were written after the lecture, but have not been carefully polished. The bibliography is not carefully put together. For an old but wonderful introduction to stable group theory, see [13]. For an update, often with results stated in great generality, see [15]. On groups of finite Morley rank, [2] gives the state of play prior to serious intrusion of methods from the classification of finite simple groups, and [1] gives closetocurrent information. 1 Motivation for stable group theory Consider an algebraically closed field K, and a linear algebraic group G(K) ≤ GLn(K). This is a subgroup of GLn(K) defined by the vanishing of some polynomials over K. Recall that the case when G(K) is simple is classified – that is, the simple algebraic groups over K are classified, via their root
A PROPERTY OF SMALL GROUPS
"... Abstract. A group is small if it has countably many pure ntypes for each integer n. It is shown that in a small group, subgroups which are definable with parameters in a finitely generated algebraic closure satisfy local descending chain conditions. An infinite small group has an infinite abelian s ..."
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Abstract. A group is small if it has countably many pure ntypes for each integer n. It is shown that in a small group, subgroups which are definable with parameters in a finitely generated algebraic closure satisfy local descending chain conditions. An infinite small group has an infinite abelian subgroup, which may not be definable. A nilpotent small group is the central product of a definable divisible group with one of bounded exponent. In a group with simple theory, any set of pairwise commuting elements is contained in a definable finitebyabelian subgroup. As a corollary, a group with small and simple theory has an infinite definable finitebyabelian subgoup. A connected group of Morley rank one is abelian [20, Reineke]. Better, a connected omegastable group of minimal Morley rank is abelian, from which it follows that every infinite omegastable group has a definable infinite abelian subgroup [6, Cherlin]. Berline and Lascar generalised this to superstable groups in [4]. More recently, Poizat introduced dminimal structures which englobe minimal ones, and proved a dminimal group to be abelianbyfinite [17]. He went further showing that an
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY
, 2010
"... The Algebraicity Conjecture treats modeltheoretic foundations of algebraic group theory. It states that any simple group of finite Morley rank is an algebraic group over an algebraically closed field. In the mid1990s a view was consolidated that this project falls into four cases of different flav ..."
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The Algebraicity Conjecture treats modeltheoretic foundations of algebraic group theory. It states that any simple group of finite Morley rank is an algebraic group over an algebraically closed field. In the mid1990s a view was consolidated that this project falls into four cases of different flavour: even type, mixed type, odd type, and degenerate type. This book contains a proof of the conjecture in the first two cases, and much more besides: insight into the current state of the other cases (which are very much open), applications for example to permutation groups of finite Morley rank, and open questions. The book will be of interest to both model theorists and group theorists: techniques from the classification of finite simple groups (CFSG), and from other aspects of group theory (e.g., black box groups in computational group theory, and the theory of Tits buildings, especially of generalised polygons) play a major role. The techniques used are primarily group theoretic, but the history and motivation are more model theoretic. The origins of the conjecture, as with much modern model theory, lie in Morley’s Theorem: this states that if T is a complete theory