Anand Pillay

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Anand Pillay is a British-American mathematician and logician known for his work in model theory, particularly in the areas of stability theory and classification theory. He has held academic positions at various universities, including the University of Notre Dame, and is recognized for his significant contributions to mathematical logic. Pillay has authored numerous research articles and books in model theory, and he is a Fellow of the American Mathematical Society.

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Geometric Stability Theory

1996-09-12

Anand Pillay

Abstract This book gives an account of the fundamental results in geometric stability theory, a subject that has grown out of categoricity and classification theory. This approach studies the fine … Abstract This book gives an account of the fundamental results in geometric stability theory, a subject that has grown out of categoricity and classification theory. This approach studies the fine structure of models of stable theories, using the geometry of forking; this often achieves global results relevant to classification theory. Topics range from Zilber-Cherlin classification of infinite locally finite homogenous geometries, to regular types, their geometries, and their role in superstable theories. The structure and existence of definable groups is featured prominently, as is work by Hrushovski. The book is unique in the range and depth of material covered and will be invaluable to anyone interested in modern model theory.

Groups, measures, and the NIP

2007-02-02

Ehud Hrushovski , Ya’acov Peterzil , Anand Pillay

We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating … We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating definably compact groups $G$ in saturated $o$-minimal structures to compact Lie groups. We also prove some other structural results about such $G$, for example the existence of a left invariant finitely additive probability measure on definable subsets of $G$. We finally introduce the new notion of "compact domination" (domination of a definable set by a compact space) and raise some new conjectures in the $o$-minimal case.

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On NIP and invariant measures

2011-05-13

Ehud Hrushovski , Anand Pillay

We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are … We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [16]. Among key results are (i) if p = \mathrm{tp}(b/A) does not fork over A then the Lascar strong type of b over A coincides with the compact strong type of b over A and any global nonforking extension of p is Borel definable over \mathrm{bdd}(A) , (ii) analogous statements for Keisler measures and definable groups, including the fact that G^{000} = G^{00} for G definably amenable, (iii) definitions, characterizations and properties of “generically stable” types and groups, (iv) uniqueness of invariant (under the group action) Keisler measures on groups with finitely satisfiable generics, (v) a proof of the compact domination conjecture for (definably compact) commutative groups in o -minimal expansions of real closed fields.

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On groups and fields definable in o-minimal structures

1988-09-01

Anand Pillay

Definable sets in ordered structures

1984-01-01

Anand Pillay , Charles Steinhorn

On introduit la notion de theorie O-minimale des structures ordonnees, une telle theorie etant telle que les sous-ensembles definissables de ses modeles soient particulierement simples On introduit la notion de theorie O-minimale des structures ordonnees, une telle theorie etant telle que les sous-ensembles definissables de ses modeles soient particulierement simples

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Definable sets in ordered structures. II

1986-01-01

Julia F. Knight , Anand Pillay , Charles Steinhorn

It is proved that any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-minimal structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> … It is proved that any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-minimal structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (in which the underlying order is dense) is strongly <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-minimal (namely, every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> elementarily equivalent to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-minimal). It is simultaneously proved that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-minimal, then every definable set of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-tuples of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has finitely many "definably connected components."

Groups definable in local fields and pseudo-finite fields

1994-02-01

Ehud Hrushovski , Anand Pillay

Coordinatisation and canonical bases in simple theories

2000-03-01

Bradd Hart , Byunghan Kim , Anand Pillay

In this paper we discuss several generalization of theorems from stability theory to simple theories. Cherlin and Hrushovski, in [2] develop a substitute for canonical bases in finite rank, ω-categorical … In this paper we discuss several generalization of theorems from stability theory to simple theories. Cherlin and Hrushovski, in [2] develop a substitute for canonical bases in finite rank, ω-categorical supersimple theories. Motivated by methods there, we prove the existence of canonical bases (in a suitable sense) for types in any simple theory. This is done in Section 2. In general these canonical bases will (as far as we know) exist only as “hyperimaginaries”, namely objects of the form a / E where a is a possibly infinite tuple and E a type-definable equivalence relation. (In the supersimple, ω-categorical case, these reduce to ordinary imaginaries.) So in Section 1 we develop the general theory of hyperimaginaries and show how first order model theory (including the theory of forking) generalises to hyperimaginaries. We go on, in Section 3 to show the existence and ubiquity of regular types in supersimple theories, ω-categorical simple structures and modularity is discussed in Section 4. It is also shown here how the general machinery of simplicity simplifies some of the general theory of smoothly approximable (or Lie-coordinatizable) structures from [2]. Throughout this paper we will work in a large, saturated model M of a complete theory T . All types, sets and sequences will have size smaller than the size of M . We will assume that the reader is familiar with the basics of forking in simple theories as laid out in [4] and [6]. For basic stability-theoretic results concerning regular types, orthogonality etc., see [1] or [9].

TYPE-DEFINABILITY, COMPACT LIE GROUPS, AND o-MINIMALITY

2004-12-01

Anand Pillay

We study type-definable subgroups of small index in definable groups, and the structure on the quotient, in first order structures. We raise some conjectures in the case where the ambient … We study type-definable subgroups of small index in definable groups, and the structure on the quotient, in first order structures. We raise some conjectures in the case where the ambient structure is o-minimal. The gist is that in this o-minimal case, any definable group G should have a smallest type-definable subgroup of bounded index, and that the quotient, when equipped with the logic topology, should be a compact Lie group of the "right" dimension. I give positive answers to the conjectures in the special cases when G is 1-dimensional, and when G is definably simple.

Jet spaces of varieties over differential and difference fields

2003-12-01

Anand Pillay , Martin Ziegler

Generically stable and smooth measures in NIP theories

2012-12-13

Ehud Hrushovski , Anand Pillay , Pierre Simon

We formulate the measure analogue of generically stable types in first order theories with $NIP$ (without the independence property), giving several characterizations, answering some questions from an earlier paper by … We formulate the measure analogue of generically stable types in first order theories with $NIP$ (without the independence property), giving several characterizations, answering some questions from an earlier paper by Hrushovski and Pillay, and giving another treatment of uniqueness results from the same paper. We introduce a notion of "generic compact domination", relating it to stationarity of the Keisler measures, and also giving definable group versions. We also prove the "approximate definability" of arbitrary Borel probability measures on definable sets in the real and $p$-adic fields.

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Model Theory with Applications to Algebra and Analysis

2008-05-22

Zoé Chatzidakis , Dugald Macpherson , Anand Pillay +1 more

The second of a two volume set showcasing current research in model theory and its connections with number theory, algebraic geometry, real analytic geometry and differential algebra. Each volume contains … The second of a two volume set showcasing current research in model theory and its connections with number theory, algebraic geometry, real analytic geometry and differential algebra. Each volume contains a series of expository essays and research papers around the subject matter of a Newton Institute Semester on Model Theory and Applications to Algebra and Analysis. The articles convey outstanding new research on topics such as model theory and conjectures around Mordell-Lang; arithmetic of differential equations, and Galois theory of difference equations; model theory and complex analytic geometry; o-minimality; model theory and non-commutative geometry; definable groups of finite dimension; Hilbert's tenth problem; and Hrushovski constructions. With contributions from so many leaders in the field, this book will undoubtedly appeal to all mathematicians with an interest in model theory and its applications, from graduate students to senior researchers and from beginners to experts.

Lovely pairs of models

2003-06-09

Itay Ben-Yaacov , Anand Pillay , Evgueni Vassiliev

On Groups and Rings Definable In O-Minimal Expansions of Real Closed Fields

1996-01-01

Margarita Otero , Ya’acov Peterzil , Anand Pillay

Let 〈R, >,+,⋅〉 be a real closed field, and let M be an o-minimal expansion of R. We prove here several results regarding rings and groups which are definable in … Let 〈R, >,+,⋅〉 be a real closed field, and let M be an o-minimal expansion of R. We prove here several results regarding rings and groups which are definable in M. We show that every M–definable ring without zero divisors is definably isomorphic to R, R(√(−l)) or the ring of quaternions over R. One corollary is that no model of Texp is interpretable in a model of Tan.

First order topological structures and theories

1987-09-01

Anand Pillay

In this paper we introduce the notion of a first order topological structure, and consider various possible conditions on the complexity of the definable sets in such a structure, drawing … In this paper we introduce the notion of a first order topological structure, and consider various possible conditions on the complexity of the definable sets in such a structure, drawing several consequences thereof. Our aim is to develop, for a restricted class of unstable theories, results analogous to those for stable theories. The “material basis” for such an endeavor is the analogy between the field of real numbers and the field of complex numbers, the former being a “nicely behaved” unstable structure and the latter the archetypal stable structure. In this sense we try here to situate our work on o -minimal structures [PS] in a general topological context. Note, however, that the p -adic numbers, and structures definable therein, will also fit into our analysis. In the remainder of this section we discuss several ways of studying topological structures model-theoretically. Eventually we fix on the notion of a structure in which the topology is “explicitly definable” in the sense of Flum and Ziegler [FZ]. In §2 we introduce the hypothesis that every definable set is a Boolean combination of definable open sets. In §3 we introduce a “dimension rank” on (closed) definable sets. In §4 we consider structures on which this rank is defined, and for which also every definable set has a finite number of definably connected definable components. We show that prime models over sets exist under such conditions.

Some foundational questions concerning differential algebraic groups

1997-05-01

Anand Pillay

In this paper we solve some problems posed by Kolchin about differential algebraic groups.The main result (from which the others follow) is the embeddability of any differential algebraic group in … In this paper we solve some problems posed by Kolchin about differential algebraic groups.The main result (from which the others follow) is the embeddability of any differential algebraic group in an algebraic group.A crucial intermediate result, and one of independent interest, is a generalisation of Weil's theorem on recovering an algebraic group from birational data, to pro-algebraic groups.

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Definable Sets in Ordered Structures. I

1986-06-01

Anand Pillay , Charles Steinhorn

This paper introduces and begins the study of a well-behaved class of linearly ordered structures, the ^-minimal structures.The definition of this class and the corresponding class of theories, the strongly … This paper introduces and begins the study of a well-behaved class of linearly ordered structures, the ^-minimal structures.The definition of this class and the corresponding class of theories, the strongly ©-minimal theories, is made in analogy with the notions from stability theory of minimal structures and strongly minimal theories.Theorems 2.1 and 2.3, respectively, provide characterizations of C-minimal ordered groups and rings.Several other simple results are collected in §3.The primary tool in the analysis of ¿¡-minimal structures is a strong analogue of "forking symmetry," given by Theorem 4.2.This result states that any (parametrically) definable unary function in an (5-minimal structure is piecewise either constant or an order-preserving or reversing bijection of intervals.The results that follow include the existence and uniqueness of prime models over sets (Theorem 5.1) and a characterization of all N0-categorical ¿¡¡-minimal structures (Theorem 6.1).

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Simple algebraic and semialgebraic groups over real closed fields

2000-06-13

Ya’acov Peterzil , Anand Pillay , Sergei Starchenko

We continue the investigation of infinite, definably simple groups which are definable in o-minimal structures. In<italic>Definably simple groups in o-minimal structures</italic>, we showed that every such group is a semialgebraic … We continue the investigation of infinite, definably simple groups which are definable in o-minimal structures. In<italic>Definably simple groups in o-minimal structures</italic>, we showed that every such group is a semialgebraic group over a real closed field. Our main result here, stated in a model theoretic language, is that every such group is either bi-interpretable with an algebraically closed field of characteristic zero (when the group is stable) or with a real closed field (when the group is unstable). It follows that every abstract isomorphism between two unstable groups as above is a composition of a semialgebraic map with a field isomorphism. We discuss connections to theorems of Freudenthal, Borel-Tits and Weisfeiler on automorphisms of real Lie groups and simple algebraic groups over real closed fields.

Definability and definable groups in simple theories

1998-09-01

Anand Pillay

Abstract We continue the study of simple theories begun in [3] and [5]. We first find the right analogue of definability of types. We then develop the theory of generic … Abstract We continue the study of simple theories begun in [3] and [5]. We first find the right analogue of definability of types. We then develop the theory of generic types and stabilizers for groups definable in simple theories. The general ideology is that the role of formulas (or definability) in stable theories is replaced by partial types (or ∞-definability) in simple theories.

Definable Sets in Ordered Structures. II

1986-06-01

Julia F. Knight , Anand Pillay , Charles Steinhorn

It is proved that any O-minimal structure M (in which the underlying order is dense) is strongly O-minimal (namely, every N elementarily equivalent to M is O-minimal).It is simultaneously proved … It is proved that any O-minimal structure M (in which the underlying order is dense) is strongly O-minimal (namely, every N elementarily equivalent to M is O-minimal).It is simultaneously proved that if M is 0minimal, then every definable set of n-tuples of M has finitely many "definably connected components."0. Introduction.In this paper we study the structure of definable sets (of tuples) in an arbitrary O-minimal structure M (in which the underlying order is dense).Recall from [PSI, PS2] that the structure M is said to be O-minimal if M = (M, <, Ri)iei, where < is a total ordering on M and every definable (with parameters) subset of M is a finite union of points in M and intervals (a, b) where aE M or a = -co and b E M or b = +co.M is said to be strongly O-minimal if every N which is elementarily equivalent to M is O-minimal.We will always assume that the underlying order of M is a dense order with no first or least element.In this paper we also introduce the notion of a definable set X C Mn being definably connected, and we prove THEOREM 0.1.Let M be O-minimal.Then any definable X C Mn is a disjoint union of finitely many definably connected definable sets.THEOREM 0.2.If M is O-minimal, then M is strongly O-minimal.THEOREM 0.3.(a) Let M be O-minimal and let (p(xi,...,xn,yi,...,ym) be any formula of L (the language for M).Then there is K < uj such that for any b E Mm, the set <j>(x,b)M (= {5 E Mn:M t= 4>(a,b)}) has at most K definably connected components.(b) If M is a O-minimal expansion o/(R, <), then in (a) we can replace definably connected by connected.Theorems 0.1 and 0.2 are proved simultaneously by a rather complicated induction argument (outlined in §3 and undertaken in § §4 and 5).Theorem 0.3 follows from Theorems 0.1 and 0.2 by a compactness argument.Let us remark that if M is the field of real numbers, or more generally any real closed field, then by Tarski's quantifier elimination [T], M is (strongly) O-minimal and moreover the definable sets (of n-tuples) in M are precisely the semialgebraic

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Additive reducts of real closed fields

1992-03-01

David Marker , Ya’acov Peterzil , Anand Pillay

In [MP] Marker and Pillay showed that if X ⊂ C n is constructible but ( C , +, X ) is not locally modular, then multiplication is definable in … In [MP] Marker and Pillay showed that if X ⊂ C n is constructible but ( C , +, X ) is not locally modular, then multiplication is definable in the structure ( C , +, X ). That result extended earlier results of Martin [M] and Rabinovich and Zil'ber [RZ]. Here we will examine additive reducts of R and Q p . Definition. A subset X of R n is called semialgebraic if it is definable in the structure ( R , +,·). A subset X of R n is called semilinear if it is definable in the structure ( R , +, <,λ r ) r∈ b , where λ r is the function x ↦ rx [scalar multiplication by r ]. Every semilinear set is a Boolean combination of sets of the form { : p ( ) = 0} and { : q ( ) > 0}, where p ( ) and q ( ) are linear polynomials. Van den Dries asked the following question: if X is semialgebraic but not semilinear, can we define multiplication in ( R , +, <, X )? This was answered negatively by Pillay, Scowcroft and Steinhorn. Theorem 1.1 [PSS]. Suppose X ⊂ R n is semialgebraic and X ⊂ I n for some bounded interval I. Then multiplication is not definable in ( R , +, <, X ,λ r ) r ∈ R . In particular if X = · ∣ [0, l ] 2 , the graph of multiplication restricted to the unit interval, then X is not semilinear so we have a negative answer to van den Dries' question. Peterzil showed that this is the only restriction.

Definable sets in ordered structures. III

1988-01-01

Anand Pillay , Charles Steinhorn

We show that any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="o"> <mml:semantics> <mml:mi>o</mml:mi> <mml:annotation encoding="application/x-tex">o</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-minimal structure has a strongly <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="o"> <mml:semantics> <mml:mi>o</mml:mi> <mml:annotation encoding="application/x-tex">o</mml:annotation> </mml:semantics> … We show that any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="o"> <mml:semantics> <mml:mi>o</mml:mi> <mml:annotation encoding="application/x-tex">o</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-minimal structure has a strongly <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="o"> <mml:semantics> <mml:mi>o</mml:mi> <mml:annotation encoding="application/x-tex">o</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-minimal theory.

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A descending chain condition for groups definable in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>o</mml:mi></mml:math>-minimal structures

2005-03-22

Alessandro Berarducci , Margarita Otero , Ya’acov Peterzil +1 more

Connected components of definable groups and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>o</mml:mi></mml:math>-minimality I

2012-06-29

Annalisa Conversano , Anand Pillay

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Generic sets in definably compact groups

2007-01-01

Ya’acov Peterzil , Anand Pillay

A subset $X$ of a group $G$ is called left generic if finitely many left translates of $X$ cover $G$. Our main result is that if $G$ is a definably … A subset $X$ of a group $G$ is called left generic if finitely many left translates of $X$ cover $G$. Our main result is that if $G$ is a definably compact group in an o-minimal structure and a definable $X\subseteq G$ is not right generic then i

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The geometry of forking and groups of finite Morley rank

1995-12-01

Anand Pillay

Abstract The notion of CM-triviality was introduced by Hrushovski, who showed that his new strongly minimal sets have this property. Recently Baudisch has shown that his new ω 1 -categorical … Abstract The notion of CM-triviality was introduced by Hrushovski, who showed that his new strongly minimal sets have this property. Recently Baudisch has shown that his new ω 1 -categorical group has this property. Here we show that any group of finite Morley rank definable in a CM-trivial theory is nilpotent-by-finite, or equivalently no simple group of finite Morley rank can be definable in a CM-trivial theory.

Closed sets and chain conditions in stable theories

1984-12-01

Anand Pillay , Gabriel Srour

An impressive theory has been developed, largely by Shelah, around the notion of a stable theory. This includes detailed structure theorems for the models of such theories as well as … An impressive theory has been developed, largely by Shelah, around the notion of a stable theory. This includes detailed structure theorems for the models of such theories as well as a generalized notion of independence. The various stability properties can be defined in terms of the numbers of types over sets, or in terms of the complexity of definable sets. In the concrete examples of stable theories, however, one finds an important distinction between “positive” and “negative” information, such a distinction not being an a priori consequence of the general definitions. In the naive examples this may take the form of distinguishing between say a class of a definable equivalence relation and the complement of a class. In the more algebraic examples, this distinction may have a “topological” significance, for example with the Zariski topology on (the set of n -tuples of) an algebraically closed field, the “closed” sets being those given by sets of polynomial equalities. Note that in the latter case, every definable set is a Boolean combination of such closed sets (the definable sets are precisely the constructible sets). Similarly, stability conditions in practice reduce to chain conditions on certain “special” definable sets (e.g. in modules, stable groups). The aim here is to develop and present such notions in the general (model-theoretic) context. The basic notion is that of an “equation”. Given a complete theory T in a language L , an L -formula φ ( x̄, ȳ ) is said to be an equation (in x̄ ) if any collection Φ of instances of φ (i.e. of formulae φ ( x̄, ā )) is equivalent to a finite subset Φ′ ⊂ Φ .

On central extensions and definably compact groups in o-minimal structures

2010-11-17

Ehud Hrushovski , Ya’acov Peterzil , Anand Pillay

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Generic stability, regularity, and quasiminimality

2011-09-07

Anand Pillay , Predrag Tanović

We study the notions generic stability, regularity, homogeneous pregeometries, quasiminimality, and their mutual relations, in arbitrary first order theories. We prove that “infinite-dimensional homogeneous pregeometries” coincide with generically stable strongly … We study the notions generic stability, regularity, homogeneous pregeometries, quasiminimality, and their mutual relations, in arbitrary first order theories. We prove that “infinite-dimensional homogeneous pregeometries” coincide with generically stable strongly regular types (p(x), x = x). We prove that in a theory without the strict order property, regular types are generically stable, and prove analogous results for quasiminimal structures. We prove that the “generic type” of a quasiminimal structure is “locally strongly regular”.

1998-01-01

Koichiro Ikeda , Akito Tsuboi , Anand Pillay

Abstract A theory T is called almost 𝒩 0 ‐categorical if for any pure types p 1 ( x 1 ),…, p n (x n ) there are only finitely … Abstract A theory T is called almost 𝒩 0 ‐categorical if for any pure types p 1 ( x 1 ),…, p n (x n ) there are only finitely many pure types which extend p 1 ( x 1 ) ∪…∪ p n (x n ) . It is shown that if T is an almost 𝒩 0 ‐categorical theory with I (𝒩 0 , T ) = 3, then a dense linear ordering is interpretable in T .

1982-03-01

Anand Pillay

We take a fixed countable model M 0 , and we look at the structure of and number of its countable elementary extensions (up to isomorphism over M 0 ). … We take a fixed countable model M 0 , and we look at the structure of and number of its countable elementary extensions (up to isomorphism over M 0 ). Assuming that S ( M 0 ) is countable, we prove that if N is a weakly minimal extension of , and if then there is an elementary embedding of N into M over M 0 ), then N is homogeneous over M 0 . Moreover the condition that ∣ S ( M 0 )∣ = ℵ 0 cannot be removed. Under the hypothesis that M 0 contains no infinite set of tuples ordered by a formula, we prove that M 0 has infinitely many countable elementary extensions up to isomorphism over M 0 . A preliminary result is that all types over M 0 are definable, and moreover is definable over M 0 if and only if is definable over M 0 (forking symmetry). We also introduce a notion of relative homogeneity, and show that a large class of elementary extensions of M 0 are relatively homogeneous over M 0 (under the assumptions that M 0 has no order and S ( M 0 ) is countable). I will now discuss the background to and motivation behind the results in this paper, and also the place of this paper relative to other conjectures and investigations. To simplify notation let T denote the complete diagram of M 0 . First, our result that if M 0 has no order then T has infinitely many countable models is related to the following conjecture: any theory with a finite number (more than one) of countable models is unstable.

A Lindemann-Weierstrass theorem for semi-abelian varieties over function fields

2009-12-02

Daniel Bertrand , Anand Pillay

We prove an analogue of the Lindemann-Weierstrass theorem (that the exponentials of a $\mathbb {Q}$-linearly independent set of algebraic numbers are algebraically independent), replacing $\mathbb {Q}^{alg}$ by $\mathbb {C}(t)^{alg}$ and … We prove an analogue of the Lindemann-Weierstrass theorem (that the exponentials of a $\mathbb {Q}$-linearly independent set of algebraic numbers are algebraically independent), replacing $\mathbb {Q}^{alg}$ by $\mathbb {C}(t)^{alg}$ and $\mathbb {G}_{m}^{n}$ by a semi-abelian variety over $\mathbb {C}(t)^{alg}$. Both the formulations of our results and the methods are differential algebraic in nature.

2000-06-01

Ehud Hrushovski , Anand Pillay

Let A be a semi-abelian variety, and X a subvariety of A , both defined over a number field. Assume that X does not contain X 1 + X 2 … Let A be a semi-abelian variety, and X a subvariety of A , both defined over a number field. Assume that X does not contain X 1 + X 2 for any positive-dimensional subvarieties X 1 , X 2 of A . Let Γ be a subgroup of A ( C ) of finite rational rank. We give doubly exponential bounds for the size of ( X ∩ Γ)\ X ( Ǭ ). Among the ingredients is a uniform bound, doubly exponential in the data, on finite sets which are quantifier-free definable in differentially closed fields. We also give uniform bounds on X ∩ Γ in the case where X contains no translate of any semi-abelian subvariety of A and Γ is a subgroup of A ( C ) of finite rational rank which has trivial intersection with A ( Ǭ ). (Here A is assumed to be defined over a number field, but X need not be.)

FORKING IN THE FREE GROUP

2007-11-16

Anand Pillay

We study model-theoretic and stability-theoretic properties of the non-abelian free group in the light of Sela's recent result on stability and results announced by Bestvina and Feighn on ‘negligible subsets' … We study model-theoretic and stability-theoretic properties of the non-abelian free group in the light of Sela's recent result on stability and results announced by Bestvina and Feighn on ‘negligible subsets' of free groups. We point out analogies between the free group and so-called bad groups of finite Morley rank, and prove ‘non-CM-triviality' of the free group.

Some model theory of compact complex spaces

2000-01-01

Anand Pillay

We make several observations about the category C of compact complex manifolds, considered as a many-sorted structure of finite Morley rank. We also point out that the Mordell-Lang conjecture holds … We make several observations about the category C of compact complex manifolds, considered as a many-sorted structure of finite Morley rank. We also point out that the Mordell-Lang conjecture holds for complex tori: if A is a complex torus, Γ a finitely generated subgroup of A, and X an analytic subvariety of A, then X ∩ Γ is a finite union of translates of subgroups of A. This is implicit in the literature but we give an elementary reduction to the abelian variety case. We discuss analogies between C and the category of finite dimensional differential algebraic sets. (A brief survey of the Mordell-Lang conjecture and model-theoretic contributions is also included.)

Automorphism groups of prime models, and invariant measures

2025-02-28

Anand Pillay

Bass modules and embeddings into free modules

2025-02-01

Anand Pillay , Philipp Rothmaler

Compactifications of pseudofinite and pseudoamenable groups

2025-01-14

Gabriel Conant , Ehud Hrushovski , Anand Pillay

We first give simplified and corrected accounts of some results in work by Pillay (2017) on compactifications of pseudofinite groups. For instance, we use a classical theorem of Turing (1938) … We first give simplified and corrected accounts of some results in work by Pillay (2017) on compactifications of pseudofinite groups. For instance, we use a classical theorem of Turing (1938) to give a simplified proof that any definable compactification of a pseudofinite group has an abelian connected component. We then discuss the relationship between Turing’s work, the Jordan–Schur theorem, and a (relatively) more recent result of Kazhdan (1982) on approximate homomorphisms, and we use this to widen our scope from finite groups to amenable groups. In particular, we develop a suitable continuous logic framework for dealing with definable homomorphisms from pseudoamenable groups to compact Lie groups. Together with the stabilizer theorems of Hrushovski (2012) and Montenegro et al. (2020), we obtain a uniform (but non-quantitative) analogue of Bogolyubov’s lemma for sets of positive measure in discrete amenable groups. We conclude with brief remarks on the case of amenable topological groups.

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Bass modules and embeddings into free modules

2025-01-07

Anand Pillay , Philipp Rothmaler

We show that the free module of infinite rank $R^{(\kappa)}$ purely embeds every $\kappa$-generated flat left $R$-module iff $R$ is left perfect. Using a Bass module corresponding to a descending … We show that the free module of infinite rank $R^{(\kappa)}$ purely embeds every $\kappa$-generated flat left $R$-module iff $R$ is left perfect. Using a Bass module corresponding to a descending chain of principal right ideals, we construct a model of the theory $T$ of $R^{(\kappa)}$ whose projectivity is equivalent to left perfectness, which allows to add a `stronger' equivalent condition: $R^{(\kappa)}$ purely embeds every $\kappa$-generated flat left $R$-module which is a model of $T$. We extend the model-theoretic construction of this Bass module to arbitrary descending chains of pp formulas, resulting in a `Bass theory' of pure-projective modules. We put this new theory to use by, among other things, reproving an old result of Daniel Simson about pure-semisimple rings and Mittag-Leffler modules. This paper is a condensed version, solely about modules, of our larger work arXiv:2407.15864, with two new results added about cyclically presented modules (Cor.14) and finitely presented cyclic modules (Rem.15).

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Open Subgroups of p-Adic Algebraic Groups

2025-01-01

Anand Pillay , Ningyuan Yao

An arithmetic algebraic regularity lemma

2024-12-15

Anand Pillay , Atticus Stonestrom

We give an 'arithmetic regularity lemma' for groups definable in finite fields, analogous to Tao's 'algebraic regularity lemma' for graphs definable in finite fields. More specifically, we show that, for … We give an 'arithmetic regularity lemma' for groups definable in finite fields, analogous to Tao's 'algebraic regularity lemma' for graphs definable in finite fields. More specifically, we show that, for any $M>0$, any finite field $\mathbf{F}$, and any definable group $(G,\cdot)$ in $\mathbf{F}$ and definable subset $D\subseteq G$, each of complexity at most $M$, there is a normal definable subgroup $H\leqslant G$, of index and complexity $O_M(1)$, such that the following holds: for any cosets $V,W$ of $H$, the bipartite graph $(V,W,xy^{-1}\in D)$ is $O_M(|\mathbf{F}|^{-1/2})$-quasirandom. Various analogous regularity conditions follow; for example, for any $g\in G$, the Fourier coefficient $||\widehat{1}_{H\cap Dg}(\pi)||_{\mathrm{op}}$ is $O_M(|\mathbf{F}|^{-1/8})$ for every non-trivial irreducible representation $\pi$ of $H$.

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Picard-Vessiot extensions, linear differential algebraic groups, and their torsors

2024-08-01

David Meretzky , Anand Pillay

Let K be a differential field with algebraically closed field of constants CK. Let KPV∞ be the (iterated) Picard-Vessiot closure of K. Let G be a linear differential algebraic group … Let K be a differential field with algebraically closed field of constants CK. Let KPV∞ be the (iterated) Picard-Vessiot closure of K. Let G be a linear differential algebraic group over K, and X a differential algebraic torsor for G over K. We prove that X(KPV∞) is Kolchin dense in X. In the special case that G is finite-dimensional we prove that X(KPV∞)=X(Kdiff) (where Kdiff is the differential closure of K). We also give close relationships between Picard-Vessiot extensions of K and the category of torsors for suitable finite-dimensional linear differential algebraic groups over K.

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Automorphism groups of prime models, and invariant measures

2024-05-20

Anand Pillay

We adapt the notion of a (relatively) definable subset of Aut(M) when M is a saturated model to the case Aut(M/A) when M is atomic and strongly omega-homogeneous over A. … We adapt the notion of a (relatively) definable subset of Aut(M) when M is a saturated model to the case Aut(M/A) when M is atomic and strongly omega-homogeneous over A. We discuss the existence and uniqueness of invariant measures on the Boolean algebra of definable subsets of Aut(M/A). For example when Th(M) is stable we have existence and uniqueness. We also discuss the compatibility of our definability notions with definable Galois cohomology and differential Galois theory.

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MORE ON GALOIS COHOMOLOGY, DEFINABILITY AND DIFFERENTIAL ALGEBRAIC GROUPS

2024-04-11

Omar León Sánchez , David Meretzky , Anand Pillay

Abstract As a continuation of the work of the third author in [5], we make further observations on the features of Galois cohomology in the general model theoretic context. We … Abstract As a continuation of the work of the third author in [5], we make further observations on the features of Galois cohomology in the general model theoretic context. We make explicit the connection between forms of definable groups and first cohomology sets with coefficients in a suitable automorphism group. We then use a method of twisting cohomology (inspired by Serre’s algebraic twisting) to describe arbitrary fibres in cohomology sequences—yielding a useful “finiteness” result on cohomology sets. Applied to the special case of differential fields and Kolchin’s constrained cohomology, we complete results from [3] by proving that the first constrained cohomology set of a differential algebraic group over a bounded, differentially large, field is countable.

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ON DEFINABLE GROUPS AND D-GROUPS IN CERTAIN FIELDS WITH A GENERIC DERIVATION

2024-01-15

Ya’acov Peterzil , Anand Pillay , Françoise Point

We continue our study from Peterzil et al. (2022, Preprint, arXiv:2208.08293) of finite-dimensional definable groups in models of the theory $T_{\partial }$, the model companion of an o-minimal ${\mathcal {L}}$-theory … We continue our study from Peterzil et al. (2022, Preprint, arXiv:2208.08293) of finite-dimensional definable groups in models of the theory $T_{\partial }$, the model companion of an o-minimal ${\mathcal {L}}$-theory T expanded by a generic derivation $\partial $ as in Fornasiero and Kaplan (2021, Journal of Mathematical Logic 21, 2150007).We generalize Buium's notion of an algebraic D-group to ${\mathcal {L}}$-definable D-groups, namely $(G,s)$, where G is an ${\mathcal {L}}$-definable group in a model of T, and $s:G\to \tau (G)$ is an ${\mathcal {L}}$-definable group section. Our main theorem says that every definable group of finite dimension in a model of $T_\partial $ is definably isomorphic to a group of the form $$ \begin{align*}(G,s)^\partial=\{g\in G:s(g)=\nabla g\},\end{align*} $$for some ${\mathcal {L}}$-definable D-group $(G,s)$ (where $\nabla (g)=(g,\partial g)$).We obtain analogous results when T is either the theory of p-adically closed fields or the theory of pseudo-finite fields of characteristic $0$.

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An analytic version of stable arithmetic regularity

2024-01-01

Gabriel Conant , Anand Pillay

We prove a structure theorem for stable functions on amenable groups, which extends the arithmetic regularity lemma for stable subsets of finite groups. Given a group $G$, a function $f\colon … We prove a structure theorem for stable functions on amenable groups, which extends the arithmetic regularity lemma for stable subsets of finite groups. Given a group $G$, a function $f\colon G\to [-1,1]$ is called stable if the binary function $f(x\cdot y)$ is stable in the sense of continuous logic. Roughly speaking, our main result says that if $G$ is amenable, then any stable function on $G$ is almost constant on all translates of a unitary Bohr set in $G$ of bounded complexity. The proof uses ingredients from topological dynamics and continuous model theory. We also discuss some applications, including a short proof of the noncommutative analogue of Bogolyubov's Lemma for amenable groups.

Galois groups of large simple fields

2023-10-21

Anand Pillay , Erik Walsberg

For Ehud Hrushovski, on For Ehud Hrushovski, on

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Continuous stable regularity

2023-10-10

Nicolas Chavarria , Gabriel Conant , Anand Pillay

Abstract We prove an analytic version of the stable graph regularity lemma by Malliaris and Shelah (Trans. Amer. Math. Soc. 366 (2014), no. 3, 1551–1585), which applies to stable functions … Abstract We prove an analytic version of the stable graph regularity lemma by Malliaris and Shelah (Trans. Amer. Math. Soc. 366 (2014), no. 3, 1551–1585), which applies to stable functions . Our methods involve continuous model theory and, in particular, results on the structure of local Keisler measures for stable continuous formulas. Along the way, we develop some basic tools around ultraproducts of metric structures and linear functionals on continuous formulas, and we also describe several concrete families of examples of stable functions.

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THORN FORKING, WEAK NORMALITY, AND THEORIES WITH SELECTORS

2023-09-11

Daniel Max Hoffmann , Anand Pillay

Abstract We discuss the role of weakly normal formulas in the theory of thorn forking, as part of a commentary on the paper [5]. We also give a counterexample to … Abstract We discuss the role of weakly normal formulas in the theory of thorn forking, as part of a commentary on the paper [5]. We also give a counterexample to Corollary 4.2 from that paper, and in the process discuss “theories with selectors.”

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ON GROUPS WITH DEFINABLE F-GENERICS DEFINABLE IN P-ADICALLY CLOSED FIELDS

2023-06-09

Anand Pillay , Ningyuan Yao

Abstract The aim of this paper is to develop the theory of groups definable in the p -adic field ${{\mathbb {Q}}_p}$ , with “definable f -generics” in the sense of … Abstract The aim of this paper is to develop the theory of groups definable in the p -adic field ${{\mathbb {Q}}_p}$ , with “definable f -generics” in the sense of an ambient saturated elementary extension of ${{\mathbb {Q}}_p}$ . We call such groups definable f-generic groups. So, by a “definable f -generic” or $dfg$ group we mean a definable group in a saturated model with a global f -generic type which is definable over a small model. In the present context the group is definable over ${{\mathbb {Q}}_p}$ , and the small model will be ${{\mathbb {Q}}_p}$ itself. The notion of a $\mathrm {dfg}$ group is dual, or rather opposite to that of an $\operatorname {\mathrm {fsg}}$ group (group with “finitely satisfiable generics”) and is a useful tool to describe the analogue of torsion-free o -minimal groups in the p -adic context. In the current paper our group will be definable over ${{\mathbb {Q}}_p}$ in an ambient saturated elementary extension $\mathbb {K}$ of ${{\mathbb {Q}}_p}$ , so as to make sense of the notions of f -generic type, etc. In this paper we will show that every definable f -generic group definable in ${{\mathbb {Q}}_p}$ is virtually isomorphic to a finite index subgroup of a trigonalizable algebraic group over ${{\mathbb {Q}}_p}$ . This is analogous to the o -minimal context, where every connected torsion-free group definable in $\mathbb {R}$ is isomorphic to a trigonalizable algebraic group [5, Lemma 3.4]. We will also show that every open definable f -generic subgroup of a definable f -generic group has finite index, and every f -generic type of a definable f -generic group is almost periodic, which gives a positive answer to the problem raised in [28] of whether f -generic types coincide with almost periodic types in the p -adic case.

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On pp-elimination and stability in a continuous setting

2023-02-17

Nicolas Chavarria , Anand Pillay

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2023-01-01

Alexey Ovchinnikov , Anand Pillay , Gleb Pogudin +1 more

The parameter identifiability problem for a dynamical system is to determine whether the parameters of the system can be found from data for the outputs of the system. Verifying whether … The parameter identifiability problem for a dynamical system is to determine whether the parameters of the system can be found from data for the outputs of the system. Verifying whether the parameters are identifiable is a necessary first step before a meaningful parameter estimation can take place. Non-identifiability occurs in practical models. To reparametrize a model to achieve identifiability is a challenge. The existing approaches have been shown to be useful for many important examples. However, these approaches are either limited to linear models and scaling parametrizations or are not guaranteed to find a reparametrization even if it exists. In the present paper, we prove that there always exists a locally identifiable model with the same input-output behaviour as the original one obtained from a given one by a partial specialization of the parameters. Furthermore, we give a sufficient observability condition for the existence of a state space transformation from the original model to the new one. Our proof is constructive and can be translated to an algorithm, which we illustrate by several examples.

Generalized locally compact models for approximate groups

2023-01-01

Krzysztof Krupiński , Anand Pillay

We give a proof of the existence of generalized definable locally compact models for arbitrary approximate subgroups via an application of topological dynamics in model theory. Our construction is simpler … We give a proof of the existence of generalized definable locally compact models for arbitrary approximate subgroups via an application of topological dynamics in model theory. Our construction is simpler and shorter than the original one obtained by Hrushovski in ``Beyond the Lascar group'', and it uses only basic model theory (mostly spaces of types and realizations of types). The main tools are Ellis groups from topological dynamics considered for suitable spaces of types. However, we need to redevelop some basic theory of topological dynamics for suitable ``locally compact flows'' in place of (compact) flows. We also prove that the generalized definable locally compact model which we constructed is universal in an appropriate category. We note that the main result yields structural information on definable generic subsets of definable groups, with a more precise structural result for generics in the universal cover of $\textrm{SL}_2(\mathbb{R})$.

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Approximate subgroups with bounded VC-dimension

2022-12-16

Gabriel Conant , Anand Pillay

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On the topological dynamics of automorphism groups: a model-theoretic perspective

2022-10-21

Krzysztof Krupiński , Anand Pillay

Abstract We give a model-theoretic treatment of the fundamental results of Kechris-Pestov-Todorčević theory in the more general context of automorphism groups of not necessarily countable structures. One of the main … Abstract We give a model-theoretic treatment of the fundamental results of Kechris-Pestov-Todorčević theory in the more general context of automorphism groups of not necessarily countable structures. One of the main points is a description of the universal ambit as a certain space of types in an expanded language. Using this, we recover results of Kechris et al. (Funct Anal 15:106–189, 2005), Moore (Fund Math 220:263–280, 2013), Ngyuen Van Thé (Fund Math 222: 19–47, 2013), in the context of automorphism groups of not necessarily countable structures, as well as Zucker (Trans Am Math Soc 368, 6715–6740, 2016).

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Gabriel Conant , Anand Pillay , C. Terry

Suppose G is a finite group and A\subseteq G is such that \{gA:g\in G\} has VC-dimension strictly less than k . We find algebraically well-structured sets in G which, up … Suppose G is a finite group and A\subseteq G is such that \{gA:g\in G\} has VC-dimension strictly less than k . We find algebraically well-structured sets in G which, up to a chosen \epsilon>0 , describe the structure of A and behave regularly with respect to translates of A . For the subclass of groups with uniformly fixed finite exponent r , these algebraic objects are normal subgroups with index bounded in terms of k , r , and \epsilon . For arbitrary groups, we use Bohr neighborhoods of bounded rank and width inside normal subgroups of bounded index. Our proofs are largely model-theoretic, and heavily rely on a structural analysis of compactifications of pseudofinite groups as inverse limits of Lie groups. The introduction of Bohr neighborhoods into the nonabelian setting uses model-theoretic methods related to the work of Breuillard, Green, and Tao [8] and Hrushovski [28] on approximate groups, as well as a result of Alekseev, Glebskiĭ, and Gordon [1] on approximate homomorphisms.

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1 Model theory and groups

2021-05-10

Anand Pillay

2021-01-01

Nicolas Chavarria , Gabriel Conant , Anand Pillay

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On pp elimination and stability in a continuous setting

2021-01-01

Nicolas Chavarria Gomez , Anand Pillay

We generalize pp elimination for modules, or more generally abelian structures, to a continuous logic environment where the abelian structure is equipped with a homomorphism to a compact (Hausdorff) group. … We generalize pp elimination for modules, or more generally abelian structures, to a continuous logic environment where the abelian structure is equipped with a homomorphism to a compact (Hausdorff) group. We conclude that the continuous logic theory of such a structure is stable.

2020-04-16

Alexey Ovchinnikov , Anand Pillay , Gleb Pogudin +1 more

Parameter identifiability is a structural property of an ODE model for recovering the values of parameters from the data (i.e., from the input and output variables). This property is a … Parameter identifiability is a structural property of an ODE model for recovering the values of parameters from the data (i.e., from the input and output variables). This property is a prerequisite for meaningful parameter identification in practice. In the presence of nonidentifiability, it is important to find all functions of the parameters that are identifiable. The existing algorithms check whether a given function of parameters is identifiable or, under the solvability condition, find all identifiable functions. Our first main result is an algorithm that computes all identifiable functions without any additional assumptions. Our second main result concerns the identifiability from multiple experiments. For this problem, we show that the set of functions identifiable from multiple experiments is what would actually be computed by input-output equation-based algorithms if the solvability condition is not fulfilled. We give an algorithm that not only finds these functions but also provides an upper bound for the number of experiments to be performed to identify these functions.

Coauthor	Papers Together
Ehud Hrushovski	27
Krzysztof Krupiński	23
Ya’acov Peterzil	16
Gabriel Conant	15
Thomas Scanlon	12
Rahim Moosa	11
Ningyuan Yao	11
Annalisa Conversano	9
Davide Penazzi	8
Charles Steinhorn	8
Alexey Ovchinnikov	7
Omar León Sánchez	7
Gleb Pogudin	7
Nicolas Chavarria	6
Daniel Palacín	6
Jakub Gismatullin	6
Daniel Bertrand	6
Élisabeth Bouscaren	6
Françoise Point	6
Franck Benoist	6
Philipp Rothmaler	5
Rizos Sklinos	5
Sergei Starchenko	5
Pierre Simon	5
Piotr Kowalski	4
Bradd Hart	4
Mike Haskel	3
Moshe Kamensky	3
Sławomir Solecki	3
Frank Olaf Wagner	3
Martin Bays	3
Alessandro Berarducci	3
Zoé Chatzidakis	3
David Marker	3
Michael Wibmer	3
Evgueni Vassiliev	3
M. Malliaris	3
David Meretzky	3
Margit Messmer	3
Erik Walsberg	3
Quentin Brouette	3
Predrag Tanović	3
Dugald Macpherson	3
Artem Chernikov	3
Byunghan Kim	2
Slavko Moconja	2
Rémi Jaoui	2
Enrique Casanovas	2
David Masser	2
Margarita Otero	2

Geometric Stability Theory

1996-09-12

Anand Pillay

On NIP and invariant measures

2011-05-13

Ehud Hrushovski , Anand Pillay

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Groups, measures, and the NIP

2007-02-02

Ehud Hrushovski , Ya’acov Peterzil , Anand Pillay

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Differential Algebraic Groups of Finite Dimension

1992-01-01

Alexandru Buium

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Hyperimaginaries and automorphism groups

2001-03-01

Daniel Lascar , Anthony L. Pillay

A hyperimaginary is an equivalence class of a type-definable equivalence relation on tuples of possibly infinite length. The notion was recently introduced in [1], mainly with reference to simple theories. … A hyperimaginary is an equivalence class of a type-definable equivalence relation on tuples of possibly infinite length. The notion was recently introduced in [1], mainly with reference to simple theories. It was pointed out there how hyperimaginaries still remain in a sense within the domain of first order logic. In this paper we are concerned with several issues: on the one hand, various levels of complexity of hyperimaginaries, and when hyperimaginaries can be reduced to simpler hyperimaginaries. On the other hand the issue of what information about hyperimaginaries in a saturated structure M can be obtained from the abstract group Aut(M) . In Section 2 we show that if T is simple and canonical bases of Lascar strong types exist in M eq then hyperimaginaries can be eliminated in favour of sequences of ordinary imaginaries. In Section 3, given a type-definable equivalence relation with a bounded number of classes, we show how the quotient space can be equipped with a certain compact topology. In Section 4 we study a certain group introduced in [5], which we call the Galois group of T , develop a Galois theory and make the connection with the ideas in Section 3. We also give some applications, making use of the structure of compact groups. One of these applications states roughly that bounded hyperimaginaries can be eliminated in favour of sequences of finitary hyperimaginaries. In Sections 3 and 4 there is some overlap with parts of Hrushovski's paper [2].

On groups and fields definable in o-minimal structures

1988-09-01

Anand Pillay

The Mordell-Lang conjecture for function fields

1996-01-01

Ehud Hrushovski

We give a proof of the geometric Mordell-Lang conjecture, in any characteristic. Our method involves a model-theoretic analysis of the kernel of Manin’s homomorphism and of a certain analog in … We give a proof of the geometric Mordell-Lang conjecture, in any characteristic. Our method involves a model-theoretic analysis of the kernel of Manin’s homomorphism and of a certain analog in characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

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Differential algebraic groups

1986-01-01

E. R. Kolchin

Differential Algebraic Groups

1972-07-01

P. J. Cassidy

An introduction to stability theory

2018-04-06

Daniel Palacín

These lecture notes are based on the first section of Pillay's book \[3] and they cover fundamental notions of stability theory such as defi nable types, forking calculus and canonical … These lecture notes are based on the first section of Pillay's book \[3] and they cover fundamental notions of stability theory such as defi nable types, forking calculus and canonical bases, as well as stable groups and homogeneous spaces. The approach followed here is originally due to Hrushovski and Pillay \[2], who presented stability from a local point of view. Throughout the notes, some general knowledge of model theory is assumed. I recommend the book of Tent and Ziegler \[4] as an introduction to model theory. Furthermore, the texts of Casanovas \[1] and Wagner \[5] may also be useful to the reader to obtain a different approach to stability theory.

Zariski Geometries

2010-02-01

Boris Zilber

We characterize the Zariski topologies over an algebraically closed field in terms of general dimension-theoretic properties. Some applications are given to complex manifold and to strongly minimal sets. We characterize the Zariski topologies over an algebraically closed field in terms of general dimension-theoretic properties. Some applications are given to complex manifold and to strongly minimal sets.

Topological dynamics of definable group actions

2009-01-04

Ludomir Newelski

Abstract We interpret the basic notions of topological dynamics in the model-theoretic setting, relating them to generic types of definable group actions and their generalizations. Abstract We interpret the basic notions of topological dynamics in the model-theoretic setting, relating them to generic types of definable group actions and their generalizations.

Measures and forking

1987-01-01

H. Jerome Keisler

Connected components of definable groups and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>o</mml:mi></mml:math>-minimality I

2012-06-29

Annalisa Conversano , Anand Pillay

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On compactifications and the topological dynamics of definable groups

2013-08-23

Jakub Gismatullin , Davide Penazzi , Anand Pillay

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Groups definable in local fields and pseudo-finite fields

1994-02-01

Ehud Hrushovski , Anand Pillay

Weakly Normal Groups

1987-01-01

U. Hrushovski , Anthony L. Pillay

Model theoretic aspects of the ellis semigroup

2011-12-05

Ludomir Newelski

Some foundational questions concerning differential algebraic groups

1997-05-01

Anand Pillay

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An introduction to forking

1979-09-01

Daniel Lascar , Bruno Poizat

The notion of forking has been introduced by Shelah, and a full treatment of it will appear in his book on stability [S1]. The principal aim of this paper is … The notion of forking has been introduced by Shelah, and a full treatment of it will appear in his book on stability [S1]. The principal aim of this paper is to show that it is an easy and natural notion. Consider some well-known examples of ℵ 0 -stable theories: vector spaces over Q , algebraically closed fields, differentially closed fields of characteristic 0; in each of these cases, we have a natural notion of independence: linear, algebraic and differential independence respectively. Forking gives a generalization of these notions. More precisely, if are subsets of some model and c a point of this model, the fact that the type of c over does not fork over means that there are no more relations of dependence between c and than there already existed between c and . In the case of the vector spaces, this means that c is in the space generated by only if it is already in the space generated by . In the case of differentially closed fields, this means that the minimal differential equations of c with coefficient respectively in and have the same order. Of course, these notions of dependence are essential for the study of the above mentioned structures. Forking is no less important for stable theories. A glance at Shelah's book will convince the reader that this is the case. What we have to do is the following. Assuming T stable and given and p a type on , we want to distinguish among the extensions of p to some of them that we shall call the nonforking extensions of p .

Generically stable and smooth measures in NIP theories

2012-12-13

Ehud Hrushovski , Anand Pillay , Pierre Simon

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Stable group theory and approximate subgroups

2011-06-15

Ehud Hrushovski

We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sum-product phenomenon. For a simple linear group <inline-formula content-type="math/mathml"> <mml:math … We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sum-product phenomenon. For a simple linear group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we show that a finite subset <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper X upper X Superscript negative 1 Baseline upper X EndAbsoluteValue slash StartAbsoluteValue upper X EndAbsoluteValue"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>X</mml:mi> <mml:msup> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">|X X ^{-1}X |/ |X|</mml:annotation> </mml:semantics> </mml:math> </inline-formula> bounded is close to a finite subgroup, or else to a subset of a proper algebraic subgroup of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also find a connection with Lie groups, and use it to obtain some consequences suggestive of topological nilpotence. Model-theoretically we prove the independence theorem and the stabilizer theorem in a general first-order setting.

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Definably amenable NIP groups

2017-12-27

Artem Chernikov , Pierre Simon

We study definably amenable NIP groups. We develop a theory of generics showing that various definitions considered previously coincide, and we study invariant measures. As applications, we characterize ergodic measures, … We study definably amenable NIP groups. We develop a theory of generics showing that various definitions considered previously coincide, and we study invariant measures. As applications, we characterize ergodic measures, give a proof of the conjecture of Petrykowski connecting existence of bounded orbits with definable amenability in the NIP case, and prove the Ellis group conjecture of Newelski and Pillay connecting the model-theoretic connected component of an NIP group with the ideal subgroup of its Ellis enveloping semigroup.

Jet spaces of varieties over differential and difference fields

2003-12-01

Anand Pillay , Martin Ziegler

Linear algebraic groups

1966-01-01

Armand Borel

Conventions and notation background material from algebraic geometry general notions associated with algebraic groups homogeneous spaces solvable groups Borel subgroups reductive groups rationality questions. Conventions and notation background material from algebraic geometry general notions associated with algebraic groups homogeneous spaces solvable groups Borel subgroups reductive groups rationality questions.

Locally modular regular types

1987-01-01

Ehud Hrushovski

Model Theory of Differential Fields

2005-12-15

This article surveys the model theory of differentially closed fields, an interesting setting where one can use model-theoretic methods to obtain algebraic information. The article concludes with one example showing … This article surveys the model theory of differentially closed fields, an interesting setting where one can use model-theoretic methods to obtain algebraic information. The article concludes with one example showing how this information can be used in diophantine applications. A differential field is a field K equipped with a derivation δ : K → K; recall that this means that, for x, y ∈ K, we have δ(x + y) = δ(x) + δ(y) and δ(xy) = x δ(y) + yδ(x). Roughly speaking, such a field is called differentially closed when it contains enough solutions of ordinary differential equations. This setting allows one to use model-theoretic methods, and particularly dimensiontheoretic ideas, to obtain interesting algebraic information. In this lecture I give a survey of the model theory of differentially closed fields, concluding with an example — Hrushovski’s proof of the Mordell–Lang conjecture in characteristic zero — showing how model-theoretic methods in this area can be used in diophantine applications. I will not give the proofs of the main theorems. Most of the material in Sections 1–3 can be found in [Marker et al. 1996], while the material in Section 4 can be found in [Hrushovski and Sokolovic ≥ 2001; Pillay 1996]. The primary reference on differential algebra is [Kolchin 1973], though the very readable [Kaplansky 1957] contains most of the basics needed here, as does the more recent [Magid 1994]. The book [Buium 1994] also contains an introduction to differential algebra and its connections to diophantine geometry. We refer the reader to these sources for references to the original literature. 1. Differentially Closed Fields Throughout this article all fields will have characteristic zero. A differential field is a field K equipped with a derivation δ : K → K. The field of constants is C = {x ∈ K : δ(x) = 0}. We will study differential fields using the language L = {+,− , · , δ, 0, 1}, the language of rings augmented by a unary function symbol δ. The theory of

Intersections in Jet Spaces and a Conjecture of S. Lang

1992-11-01

Alexandru Buium

Linear Algebraic Groups

1991-01-01

Armand Borel

G-compactness and groups

2008-07-04

Jakub Gismatullin , Ludomir Newelski

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Definable Compactness and Definable Subgroups of o-Minimal Groups

1999-06-01

Ya’acov Peterzil , Charles Steinhorn

The paper introduces the notion of definable compactness and within the context of o-minimal structures proves several topological properties of definably compact spaces. In particular a definable set in an … The paper introduces the notion of definable compactness and within the context of o-minimal structures proves several topological properties of definably compact spaces. In particular a definable set in an o-minimal structure is definably compact (with respect to the subspace topology) if and only if it is closed and bounded. Definable compactness is then applied to the study of groups and rings in o-minimal structures. The main result proved is that any infinite definable group in an o-minimal structure that is not definably compact contains a definable torsion-free subgroup of dimension 1. With this theorem, a complete characterization is given of all rings without zero divisors that are definable in o-minimal structures. The paper concludes with several examples illustrating some limitations on extending the theorem.

Model theory of difference fields

1999-04-08

Zoé Chatzidakis , Ehud Hrushovski

A difference field is a field with a distinguished automorphism<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"><mml:semantics><mml:mi>σ</mml:mi><mml:annotation encoding="application/x-tex">\sigma</mml:annotation></mml:semantics></mml:math></inline-formula>. This paper studies the model theory of existentially closed difference fields. We introduce a dimension theory … A difference field is a field with a distinguished automorphism<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma"><mml:semantics><mml:mi>σ</mml:mi><mml:annotation encoding="application/x-tex">\sigma</mml:annotation></mml:semantics></mml:math></inline-formula>. This paper studies the model theory of existentially closed difference fields. We introduce a dimension theory on formulas, and in particular on difference equations. We show that an arbitrary formula may be reduced into one-dimensional ones, and analyze the possible internal structures on the one-dimensional formulas when the characteristic is<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"><mml:semantics><mml:mn>0</mml:mn><mml:annotation encoding="application/x-tex">0</mml:annotation></mml:semantics></mml:math></inline-formula>.

On definable subsets of p-adic fields

1976-09-01

Angus Macintyre

The brilliant work of Ax-Kochen [1], [2], [3] and Ersov [6], and later work by Kochen [7] have made clear very striking resemblances between real closed fields and p -adically … The brilliant work of Ax-Kochen [1], [2], [3] and Ersov [6], and later work by Kochen [7] have made clear very striking resemblances between real closed fields and p -adically closed fields, from the model-theoretical point of view. Cohen [5], from a standpoint less model-theoretic, also contributed much to this analogy. In this paper we shall point out a feature of all the above treatments which obscures one important resemblance between real and p -adic fields. We shall outline a new treatment of the p -adic case (not far removed from the classical treatments cited above), and establish an new analogy between real closed and p -adically closed fields. We want to describe the definable subsets of p -adically closed fields. Tarski [9] in his pioneering work described the first-order definable subsets of real closed fields. Namely, if K is a real-closed field and X is a subset of K first-order definable on K using parameters from K then X is a finite union of nonoverlapping intervals (open, closed, half-open, empty or all of K ). In particular, if X is infinite, X has nonempty interior. Now, there is an analogous question for p -adically closed fields. If K is p -adically closed, what are the definable subsets of K ? To the best of our knowledge, this question has not been answered until now. What is the difference between the two cases? Tarski's analysis rests on elimination of quantifiers for real closed fields. Elimination of quantifiers for p -adically closed fields has been achieved [3], but only when we take a cross-section π as part of our basic data. The problem is that in the presence of π it becomes very difficult to figure out what sort of set is definable by a quantifier free formula. We shall see later that use of the cross-section increases the class of definable sets.

The classification of the semisimple differential algebraic groups and the linear semisimple differential algebraic Lie algebras

1989-02-01

Phyllis J. Cassidy

FORKING IN THE FREE GROUP

2007-11-16

Anand Pillay

Minimal bounded index subgroup for dependent theories

2007-11-09

Saharon Shelah

For a dependent theory <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper C Subscript upper T"> <mml:semantics> <mml:msub> … For a dependent theory <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper C Subscript upper T"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">C</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>T</mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">{\mathfrak {C}}_{T}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for every type definable group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the intersection of type definable subgroups with bounded index is a type definable subgroup with bounded index.

A survey of basic stability theory, with particular emphasis on orthogonality and regular types

1984-09-01

Michael Makkai

Definable sets in ordered structures

1984-01-01

Anand Pillay , Charles Steinhorn

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