Anthony L. Pillay

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All published works (28)

Psychology and COVID-19: impacts, themes and way forward

2020-06-01

COVID-19 has brought a new set of challenges at a time when poorer nations were struggling with existing burdens. However, the lockdown restrictions aimed at slowing the infection rate has … COVID-19 has brought a new set of challenges at a time when poorer nations were struggling with existing burdens. However, the lockdown restrictions aimed at slowing the infection rate has created problems of their own such as increased unemployment, poverty, and mental health problems. While the lockdown approach may be effective for public health, there is concern about the way it is formulated, the empirical basis of some restrictions, and societal impacts. There is additional concern that COVID-19 and associated restrictions disproportionately affect marginalised groups. As a discipline primarily concerned with human behaviour, Psychology has much to contribute to addressing the pandemic.

A group version of stable regularity

2018-10-24

Gabriel Conant , Anthony L. Pillay , C. Terry

Abstract We prove that, given ε > 0 and k ≥ 1, there is an integer n such that the following holds. Suppose G is a finite group and A … Abstract We prove that, given ε > 0 and k ≥ 1, there is an integer n such that the following holds. Suppose G is a finite group and A ⊆ G is k -stable. Then there is a normal subgroup H ≤ G of index at most n , and a set Y ⊆ G , which is a union of cosets of H , such that | A △ Y | ≤ε| H |. It follows that, for any coset C of H , either | C ∩ A |≤ ε| H | or | C \ A | ≤ ε | H |. This qualitatively generalises recent work of Terry and Wolf on vector spaces over $\mathbb{F}_p$ .

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Structure and regularity for subsets of groups with finite VC-dimension

2018-01-01

Gabriel Conant , Anthony L. 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 to … 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 and Hrushovski on approximate groups, as well as a result of Alekseev, Glebskii, and Gordon on approximate homomorphisms.

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Relative Manin–Mumford for Semi-Abelian Surfaces

2016-01-18

Daniel Bertrand , David Masser , Anthony L. Pillay +1 more

Abstract We show that Ribet sections are the only obstruction to the validity of the relative Manin–Mumford conjecture for one-dimensional families of semi-abelian surfaces. Applications include special cases of the … Abstract We show that Ribet sections are the only obstruction to the validity of the relative Manin–Mumford conjecture for one-dimensional families of semi-abelian surfaces. Applications include special cases of the Zilber–Pink conjecture for curves in a mixed Shimura variety of dimension 4, as well as the study of polynomial Pell equations with non-separable discriminants.

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GROUP COVERS, o-MINIMALITY, AND CATEGORICITY

2010-12-01

Alessandro Berarducci , Ya’acov Peterzil , Anthony L. Pillay

We study the model theory of "covers" of groups H definable in an o-minimal structure M. We pose the question of whether any finite central extension G of H is … We study the model theory of "covers" of groups H definable in an o-minimal structure M. We pose the question of whether any finite central extension G of H is interpretable in M, proving some cases (such as when H is abelian) as well as stating various equivalences. When M is an o-minimal expansion of the reals (so H is a definable Lie group) this is related to Milnor's conjecture [15], and many cases are known. We also prove a strong relative Lω1, ω-categoricity theorem for universal covers of definable Lie groups, and point out some notable differences with the case of covers of complex algebraic groups (studied by Zilber and his students).

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Definable groups and compact <i>p</i> -adic Lie groups

2008-05-19

Alf Onshuus , Anthony L. Pillay

We formulate p-adic analogues of the o-minimal group conjectures from the works of Hrushovski, Peterzil and Pillay [J. Amer. Math. Soc., to appear] and Pillay [J. Math. Log. 4 (2004) … We formulate p-adic analogues of the o-minimal group conjectures from the works of Hrushovski, Peterzil and Pillay [J. Amer. Math. Soc., to appear] and Pillay [J. Math. Log. 4 (2004) 147–162]; that is, we formulate versions that are appropriate for groups G definable in (saturated) P-minimal fields. We then restrict our attention to saturated models K of Th(ℚp) and Th(ℚp, an), record some elementary observations when G is defined over the standard model ℚp, and then make a detailed analysis of the case where G = E(K) for E an elliptic curve over K. Essentially, our P-minimal conjectures hold in these contexts and, moreover, our case study of elliptic curves yields counterexamples to a more naive direct translation of the o-minimal conjectures.

Linear Groups Definable in o-Minimal Structures

2002-01-01

Ya’acov Peterzil , Anthony L. Pillay , Sergei Starchenko

GALOIS GROUPS OF FIRST ORDER THEORIES

2001-11-01

Enrique Casanovas , Daniel Lascar , Anthony L. Pillay +1 more

We study the groups Gal L (T) and Gal KP (T), and the associated equivalence relations E L and E KP , attached to a first order theory T. An … We study the groups Gal L (T) and Gal KP (T), and the associated equivalence relations E L and E KP , attached to a first order theory T. An example is given where E L ≠ E KP (a non G-compact theory). It is proved that E KP is the composition of E L and the closure of E L . Other examples are given showing this is best possible.

A note on groups definable in difference fields

2001-05-22

Piotr Kowalski , Anthony L. Pillay

We prove that a group definable in a model of $ACFA$ is virtually definably embeddable in an algebraic group. We give an improved proof of the same result for groups … We prove that a group definable in a model of $ACFA$ is virtually definably embeddable in an algebraic group. We give an improved proof of the same result for groups definable in differentially closed fields. We also extend to the difference field context results on the unipotence of definable groups on affine spaces.

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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].

Definably simple groups in o-minimal structures

2000-02-24

Ya’acov Peterzil , Anthony L. Pillay , Sergei Starchenko

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G equals mathematical left-angle upper G comma dot mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G equals mathematical left-angle upper G comma dot mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mo>⋅</mml:mo> <mml:mo fence="false" stretchy="false">⟩</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}=\langle G, \cdot \rangle</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a group definable in an o-minimal structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. A subset <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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> is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-definable if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is definable in the structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mathematical left-angle upper G comma dot mathematical right-angle"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">⟨</mml:mo> <mml:mi>G</mml:mi> <mml:mo>,</mml:mo> <mml:mo>⋅</mml:mo> <mml:mo fence="false" stretchy="false">⟩</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\langle G,\cdot \rangle</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (while <italic>definable</italic> means definable in the structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper M"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">M</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>). Assume <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has no <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-definable proper subgroup of finite index. In this paper we prove that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has no nontrivial abelian normal subgroup, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the direct product of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper G"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">G</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {G}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-definable subgroups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H 1 comma ellipsis comma upper H Subscript k Baseline"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">H_1,\ldots ,H_k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that each <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Subscript i"> <mml:semantics> <mml:msub> <mml:mi>H</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">H_i</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is definably isomorphic to a semialgebraic linear group over a definable real closed field. As a corollary we obtain an o-minimal analogue of Cherlin’s conjecture.

Review: Ehud Hrushovski, Boris Zilber, Zariski Geometries

1999-06-01

Anthony L. Pillay

Ehud Hrushovski and Boris Zilber. Zariski geometries, Journal of the American Mathematical Society, vol. 9 (1996), pp. 1–56.

1999-06-01

Anthony L. Pillay

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Generic structures and simple theories

1998-11-01

Zoé Chatzidakis , Anthony L. Pillay

Supersimple fields and division rings

1998-01-01

Anthony L. Pillay , Thomas Scanlon , Frank Olaf Wagner

It is proved that any supersimple field has trivial Brauer group, and more generally that any supersimple division ring is commutative. As prerequisites we prove several results about generic types … It is proved that any supersimple field has trivial Brauer group, and more generally that any supersimple division ring is commutative. As prerequisites we prove several results about generic types in groups and fields whose theory is simple.

1-based theories ? the main gap fora-models

1995-10-01

Bradd Hart , Anthony L. Pillay , Sergei Starchenko

Triviality, NDOP and stable varieties

1993-07-01

Bradd Hart , Anthony L. Pillay , Sergei Starchenko

Tiny models of categorical theories

1992-11-01

M. Laskowski , Anthony L. Pillay , Ph. Rothmaler

Superstable differential fields

1992-03-01

Anthony L. Pillay , Zeljka Sokolovic

In this paper we study differential fields of characteristic 0 (with perhaps additional structure) whose theory is superstable. Our main result is that such a differential field has no proper … In this paper we study differential fields of characteristic 0 (with perhaps additional structure) whose theory is superstable. Our main result is that such a differential field has no proper strongly normal extensions in the sense of Kolchin [K1]. This is an approximation to the conjecture that a superstable differential field is differentially closed (although we believe the full conjecture to be false). Our result improves earlier work of Michaux [Mi] who proved that a (plain) differential field with quantifier elimination has no proper Picard-Vessiot extension. Our result is a generalisation of Michaux's, due to the fact that any plain differential field K with quantifier elimination is ω-stable. (Any quantifier free type over K defines a unique type over K in the sense of dc( k ), the differential closure of K , and as we mention below the theory of differentially closed fields is ω-stable.) The proof of our main result depends on (i) Kolchin's theory [K3] which states that any strongly normal extension L of an algebraically closed differential field K is generated over K by an element η of some algebraic group G defined over C K , the constants of K , where η satisfies some specific differential equations over K related to invariant differential forms on G (η is “ G -primitive” over K ), and (ii) the fact that a superstable field has a unique generic type which is semiregular.

Groups of dimension two and three over o-minimal structures

1991-09-01

Ali Nesin , Anthony L. Pillay , V. Razenj

Reducts of (C, +, ·) which contain +

1990-09-01

David R. Marker , Anthony L. Pillay

Abstract We show that the structure (C, +, ·) has no proper non locally modular reducts which contain +. In other words, if X ⊂ C n is constructible and … Abstract We show that the structure (C, +, ·) has no proper non locally modular reducts which contain +. In other words, if X ⊂ C n is constructible and not definable in the module structure (C, +, λ a ) a Є C (where λ a denotes multiplication by a ) then multiplication is definable in ( C , +, X ).

Semisimple stable and superstable groups

1989-12-01

John T. Baldwin , Anthony L. Pillay

A note on subgroups of the automorphism group of a saturated model, and regular types

1989-09-01

Anthony L. Pillay

Abstract Let M be a saturated model of a superstable theory and let G = Aut( M ). We study subgroups H of G which contain G (A) , A … Abstract Let M be a saturated model of a superstable theory and let G = Aut( M ). We study subgroups H of G which contain G (A) , A the algebraic closure of a finite set, generalizing results of Lascar [L] as well as giving an alternative characterization of the simple superstable theories of [P]. We also make some observations about good, locally modular regular types p in the context of p -simple types.

Weakly Normal Groups

1987-01-01

U. Hrushovski , Anthony L. Pillay

Weakly Definable Types

1986-06-01

Laurence Kirby , Anthony L. Pillay

We study some generalizations of the notion of a definable type, first in an abstract setting in terms of ultrafilters on certain Boolean algebras, and then as applied to model … We study some generalizations of the notion of a definable type, first in an abstract setting in terms of ultrafilters on certain Boolean algebras, and then as applied to model theory.

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Weakly definable types

1986-01-01

Laurence Kirby , Anthony L. Pillay

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Instability and theories with few models

1980-11-01

Anthony L. Pillay

Some results are obtained concerning <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n left-parenthesis upper T right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">n(T)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the number … Some results are obtained concerning <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n left-parenthesis upper T right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">n(T)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the number of countable models up to isomorphism, of a countable complete first order theory <italic>T</italic>. It is first proved that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n left-parenthesis upper T right-parenthesis equals 3"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n(T) = 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <italic>T</italic> has a tight prime model, then <italic>T</italic> is unstable. Secondly, it is proved that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n left-parenthesis upper T right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">n(T)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is finite and more than one, and <italic>T</italic> has few links, then <italic>T</italic> is unstable. Lastly we show that if <italic>T</italic> has an algebraic model and has few links, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n left-parenthesis upper T right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">n(T)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is infinite.

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ℵ<sub>0</sub> -Categoricity in Regular Subsets of Lω1ω

1980-10-01

Anthony L. Pillay

Common Coauthors

Coauthor	Papers Together
Sergei Starchenko	4
Ya’acov Peterzil	3
C. Terry	2
Gabriel Conant	2
Daniel Lascar	2
Bradd Hart	2
Laurence Kirby	2
U. Hrushovski	1
David Masser	1
David R. Marker	1
Enrique Casanovas	1
Alessandro Berarducci	1
V. Razenj	1
Ali Nesin	1
Piotr Kowalski	1
Alf Onshuus	1
Martin Ziegler	1
Daniel Bertrand	1
Ph. Rothmaler	1
Zoé Chatzidakis	1
Frank Olaf Wagner	1
Umberto Zannier	1
Thomas Scanlon	1
John T. Baldwin	1
Zeljka Sokolovic	1
M. Laskowski	1
Brendon Barnes	1

Commonly Cited References

On groups and fields definable in o-minimal structures

1988-09-01

Anand Pillay

Superstable groups

1986-01-01

Ch. Berline , Daniel Lascar

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.

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.

Ranks and definability in superstable theories

1976-03-01

Daniel Lascar

Weakly Normal Groups

1987-01-01

U. Hrushovski , Anthony L. Pillay

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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An application of model theory to real and p-adic algebraic groups

1989-10-01

Anand Pillay

The geometry of weakly minimal types

1985-12-01

Steven Buechler

Abstract Let T be superstable. We say a type p is weakly minimal if R ( p, L , ∞) = 1. Let M ⊨ T be uncountable and saturated, … Abstract Let T be superstable. We say a type p is weakly minimal if R ( p, L , ∞) = 1. Let M ⊨ T be uncountable and saturated, H = p ( M ). We say D ⊂ H is locally modular if for all X, Y ⊂ D with X = acl( X ) ∩ D, Y = acl( Y ) ∩ D and X ∩ Y ≠ ∅, Theorem 1. Let p ∈ S(A) be weakly minimal and D the realizations of stp( a/A) for some a realizing p. Then D is locally modular or p has Morley rank 1. Theorem 2. Let H, G be definable over some finite A, weakly minimal, locally modular and nonorthogonal. Then for all a ∈ H ∖acl( A ), b ∈ G ∖acl( A ) there are a ′ ∈ H , b ′ ∈ G such that a′ ∈ acl( abb ′ A )∖acl( aA). Similarly when H and G are the realizations of complete types or strong types over A .

Hyperimaginaries and automorphism groups

2001-03-01

Daniel Lascar , Anthony L. Pillay

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.

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>.

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 .

Regularity lemmas for stable graphs

2013-08-29

M. Malliaris , Saharon Shelah

We develop a framework in which Szemerédi's celebrated Regularity Lemma for graphs interacts with core model-theoretic ideas and techniques. Our work relies on a coincidence of ideas from model theory … We develop a framework in which Szemerédi's celebrated Regularity Lemma for graphs interacts with core model-theoretic ideas and techniques. Our work relies on a coincidence of ideas from model theory and graph theory: arbitrarily large half-graphs coincide with model-theoretic instability, so in their absence, structure theorems and technology from stability theory apply. In one direction, we address a problem from the classical Szemerédi theory. It was known that the "irregular pairs" in the statement of Szemerédi's Regularity Lemma cannot be eliminated, due to the counterexample of half-graphs (i.e., the order property, corresponding to model-theoretic instability). We show that half-graphs are the only essential difficulty, by giving a much stronger version of Szemerédi's Regularity Lemma for models of stable theories of graphs (i.e. graphs with the non-$k_*$-order property), in which there are no irregular pairs, the bounds are significantly improved, and each component satisfies an indivisibility condition. In the other direction, we take a more model-theoretic approach, and give several new Szemerédi-type partition theorems for models of stable theories of graphs. The first theorem gives a partition of any such graph into indiscernible components, meaning here that each component is either a complete or an empty graph, whose interaction is strongly uniform. This relies on a finitary version of the classic model-theoretic fact that stable theories admit large sets of indiscernibles, by showing that in models of stable theories of graphs one can extract much larger indiscernible sets than expected by Ramsey's theorem. The second and third theorems allow for a much smaller number of components at the cost of weakening the "indivisibility" condition on the components. We also discuss some extensions to graphs without the independence property. All graphs are finite and all partitions are equitable, i.e. the sizes of the components differ by at most 1. In the last three theorems, the number of components depends on the size of the graph; in the first theorem quoted, this number is a function of $\epsilon$ only as in the usual Szemerédi Regularity Lemma.

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Ultrafilters and types on models of arithmetic

1984-11-01

Laurence Kirby

A note on Lascar strong types in simple theories

1998-09-01

Byunghan Kim

Abstract Let T be a countable, small simple theory. In this paper, we prove that for such T , the notion of Lascar strong type coincides with the notion of … Abstract Let T be a countable, small simple theory. In this paper, we prove that for such T , the notion of Lascar strong type coincides with the notion of strong type, over an arbitrary set.

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Lie Groups and Algebraic Groups

1990-01-01

A. L. Onishchik , É. B. Vinberg

Some remarks on definable equivalence relations in O-minimal structures

1986-09-01

Anand Pillay

Let M be an O-minimal structure. We use our understanding, acquired in [KPS], of the structure of definable sets of n -tuples in M , to study definable (in M … Let M be an O-minimal structure. We use our understanding, acquired in [KPS], of the structure of definable sets of n -tuples in M , to study definable (in M ) equivalence relations on M n . In particular, we show that if E is an A -definable equivalence relation on M n ( A ⊂ M ) then E has only finitely many classes with nonempty interior in M n , each such class being moreover also A -definable. As a consequence, we are able to give some conditions under which an O -minimal theory T eliminates imaginaries (in the sense of Poizat [P]). If L is a first order language and M an L -structure, then by a definable set in M , we mean something of the form X ⊂ M n , n ≥ 1, where X = {( a 1 …, a n ) ∈ M n : M ⊨ ϕ (ā)} for some formula ∈ L ( M ). (Here L ( M ) means L together with names for the elements of M .) If the parameters from come from a subset A of M , we say that X is A-definable . M is said to be O-minimal if M = ( M , <,…), where < is a dense linear order with no first or last element, and every definable set X ⊂ M is a finite union of points, and intervals ( a, b ) (where a, b ∈ M ∪ {± ∞}). (This notion is as in [PS] except here we demand the underlying order be dense.) The complete theory T is said to be O-minimal if every model of T is O-minimal. (Note that in [KPS] it is proved that if M is O-minimal, then T = Th( M ) is O-minimal.) In the remainder of this section and in §2, M will denote a fixed but arbitrary O-minimal structure. A,B,C,… will denote subsets of M .

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.

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

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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Model theory of difference fields

2017-03-29

Zoé Chatzidakis

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content. A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

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Groups in simple theories

1997-01-01

Frank Olaf Wagner

Groups definable in simple theories retain the chain conditions and decomposition properties known from stable groups, up to commensurability. In the small case, if a generic type of G is … Groups definable in simple theories retain the chain conditions and decomposition properties known from stable groups, up to commensurability. In the small case, if a generic type of G is not foreign to some type q, there is a q-internal quotient. In the supersimple case, the Berline-Lascar decomposition works. One-based simple groups are finite-by-abelian-by-finite.

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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.

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.

On the category of models of a complete theory

1982-06-01

Daniel Lascar

Let T be a countable complete theory and C ( T ) the category whose objects are the models of T and morphisms are the elementary maps. The main object … Let T be a countable complete theory and C ( T ) the category whose objects are the models of T and morphisms are the elementary maps. The main object of this paper will be the study of C ( T ). The idea that a better understanding of the category may give us model theoretic information about T is quite natural: The (semi) group of automorphisms (endomorphisms) of a given structure is often a powerful tool for studying this structure. But certainly, one of the very first questions to be answered is: “to what extent does this category C ( T ) determine T ?” There is some obvious limitation: for example let T 0 be the theory of infinite sets (in a language containing only =) and T 1 the theory, in the language ( =, U ( ν 0 ), f ( ν 0 )) stating that: (1) U is infinite. (2) f is a bijective map from U onto its complement. It is quite easy to see that C ( T 0 ) is equivalent to C ( T 1 ). But, in this case, T 0 and T 1 can be “interpreted” each in the other. To make this notion of interpretation precise, we shall associate with each theory T a category, loosely denoted by T , defined as follows: (1) The objects are the formulas in the given language. (2) The morphisms from into are the formulas such that (i.e. f defines a map from ϕ into ϕ ; two morphisms defining the same map in all models of T should be identified).

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.

Remarks on some theorems of Keisler

1969-01-01

Per Lindström

This paper is an appendix to Keisler's papers [3], [4]. We assume that the reader is fully acquainted with these papers and use their terminology without further explanation. One of … This paper is an appendix to Keisler's papers [3], [4]. We assume that the reader is fully acquainted with these papers and use their terminology without further explanation. One of the main tools in a number of proofs in [4] is the following result which is all but stated explicitly in [4].

Locally modular regular types

1987-01-01

Ehud Hrushovski

Superstable fields and groups

1980-08-01

Gregory Cherlin , Saharon Shelah

Groups definable in local fields and pseudo-finite fields

1994-02-01

Ehud Hrushovski , Anand Pillay

Sous-groupes définissables d'un groupe stable

1981-03-01

Bruno Poizat

On sait, depuis les travaux de Zil'ber et de Cherlin, que le degré de Morley de la théorie T d'un groupe G totalement transcendant est l'indice du plus petit sous-groupe … On sait, depuis les travaux de Zil'ber et de Cherlin, que le degré de Morley de la théorie T d'un groupe G totalement transcendant est l'indice du plus petit sous-groupe définissable d'indice fini de G . Il est clair qu'il lui est supérieur, et l'inégalité inverse peut s'obtenir de la manière suivante: on fait agir G sur les types de S 1 ( G ) de rang de Morley maximum en associant à p , type de x au-dessus de G , le type ap de ax au-dessus de G ; on montre alors que cette action est définissable, que le fait que ap = q équivaut au fait que a satisfasse une certaine formule à paramètres dans G , ce qui est bien facile si on n'oublie pas que dans une théorie stable tous les types sont définissables; on montre ensuite que cette action est transitive, que si p et q sont de rang de Morley maximum il existe a dans G tel que ap = q , et la méthode la plus rapide, mais qui est aussi la plus sophistiquée, est d'utiliser l'argument de symétrie de la déviation employé dans la preuve de la Proposition 1 de présent article; on conclut alors puisque le degré de Morley, qui est par définition le nombre de types de rang de Morley maximum, ést egal à l'indice du stabilisateur de p , qui est définissable. Ce comportement des types de rang de Morely maximum se retrouve sans peine, si G est seulement superstable, dans celui des types de “plus petit rang continu” (encore appellé “degré de Shelah”) maximum. Pour trouver ce qui leur correspond dans le cas où G est seulement stable, il faut être un peu plus soigneux, et considérer les types p de S 1 ( G ), où G aura éventuellement été remplacé par une extension élémentaire suffisament saturée, tels que pour tout a de G ap ne dévie pas sur ∅: on montre qu'ils existent, et qu'ils sont tous conjugués par action de G ; le fait que ap = q s'exprimera cette fois par une infinité de formules et non plus par une seule.

Models and types of Peano's arithmetic

1976-03-01

Haim Gaifman

Uncountable Saturated Structures have the Small Index Property

1993-03-01

Daniel Lascar , Saharon Shelah

We prove the following theorem. Let m be an uncountable saturated structure of cardinality λ = λ<λ and assume that G is a subgroup of Aut (m) whose index is … We prove the following theorem. Let m be an uncountable saturated structure of cardinality λ = λ<λ and assume that G is a subgroup of Aut (m) whose index is less than or equal to λ. Then there exists a subset A of cardinality strictly less than λ such that every automorphism of m leaving A pointwise fixed is in G.

Advanced Topics in the Arithmetic of Elliptic Curves

1994-01-01

Joseph H. Silverman

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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Linear Algebraic Groups

1998-01-01

T. A. Springer

The first edition of this book presented the theory of linear algebraic groups over an algebraically closed field. The second edition, thoroughly revised and expanded, extends the theory over arbitrar The first edition of this book presented the theory of linear algebraic groups over an algebraically closed field. The second edition, thoroughly revised and expanded, extends the theory over arbitrar

Elliptic Functions and Transcendence

1975-01-01

David Masser

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Model theory and modules

2003-01-01

Mike Prest

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On the homology of Lie groups made discrete

1983-12-01

John Milnor

Here the homology of BG ~ is just the usual Eilenberg-MacLane homology of the uncountably infinite discrete group G 8. These homology groups are of interest in algebraic K-theory (see … Here the homology of BG ~ is just the usual Eilenberg-MacLane homology of the uncountably infinite discrete group G 8. These homology groups are of interest in algebraic K-theory (see for example Quillen), in the study of bundles with flat connection (Milnor, 1958), in the theory of foliations (Haefliger, 1973), and also in the study of scissors congruence of polyhedra (Dupont and Sah). They are difficult to compute, and tend to be rather wild. For example if G is non-trivial and connected, then Sah and Wagoner show that HE(BG~;Z) maps onto an uncountable rational vector space. (See also Harris.) The homology and cohomology groups of BG, on the other hand, are much better behaved and better understood. (Borel, 1953.) In w we will see that this Isomorphism Conjecture is true whenever the component of the identity in G is solvable. If it is true for simply-connected simple groups, then it is true for all Lie groups. It is always true for 1-dimensional homology, and is true in a number of interesting special cases for 2-dimensional homology. (See w For higher dimensional computations which tend to support the conjecture, see Karoubi, p. 256, Parry and Sah, as well as Thomason. Another partial result is the following (w If G has only finitely many components, then for any finite coefficient group A the homomorphism H.(BGS; A) --~ H,(BG; A) is split surjective. Thus we obtain a direct sum decomposition

The structure of approximate groups

2012-10-19

Emmanuel Breuillard , Ben Green , Terence Tao

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Groupes algébriques et corps de classes

1997-10-21

Jean Pierre Serre

Two conjectures regarding the stability of ω-categorical theories

1974-01-01

A. H. Lachlan

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The Arithmetic of Elliptic Curves

1986-01-01

Joseph H. Silverman

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On Triples in Arithmetic Progression

1999-12-01

Jean Bourgain

Geometric categories and o-minimal structures

1996-08-01

Lou van den Dries , Chris Miller

Groupes stables, avec types génériques réguliers

1983-06-01

Bruno Poizat

Cet article ne contient rien de révolutionnaire, et son auteur a conscience du risque qu'il court de le voir dépassé, lors de sa parution, par des travaux plus profonds. Il … Cet article ne contient rien de révolutionnaire, et son auteur a conscience du risque qu'il court de le voir dépassé, lors de sa parution, par des travaux plus profonds. Il a pensé qu'il n'était pas inutile de publier, faute de mieux, les résultats simples auxquels il est parvenu (le lecteur lui saura gré de ce caractère reposant), en espérant qu'il passeront à la postérité sous forme d'exercices dans les manuels futurs où les petits enfants apprendront la théorie des modèles. Il s'agit de groupes stables; on sait depuis longtemps que la stabilité de la théorie d'un groupe impose des “conditions de chaîne” sur ses sous-groupes définissables : l'exploitation de ce phénomène, de nature bien algébrique, qui est brièvement exposé dans la première section de cet article, a fait le bonheur d'une génération de théoriciens des modèles. Un autre phénomène, plus subtil, semble ne pas avoir épuisé sa substance : c'est celui, dont l'apparition remonte aux travaux de B. Zilber, qui est décrit dans [7] sous le nom de “types de strate maximum”, et dans [2] sous le nom de “large sets”; la deuxième section lui est consacrée: elle a été écrite dans le désir de montrer l'équivalence de ces deux approches, et aussi par repentir de n'avoir pas suffisament éclairé ses motivations dans [7]. Ce souci d'unification a eu une influence, que j'espère salutaire, sur le vocabulaire: je parle ici de “types génériques” et d'“ensembles (ou de formules) génériques”. Ces termes, ainsi que celui de “composante connexe”, sont empruntés au langage de la géométrie: un groupe algébrique, étant définissable dans la théorie d'un corps algébriquement clos, est stable, et ses “types génériques” sont les “points génériques” des géomètres.

Autour d'une conjecture de Serge Lang

1988-10-01

Marc Hindry

Algebraic and Abstract Simple Groups

1964-09-01

Jacques Tits

An exposition of Shelah's ``main gap'': counting uncountable models of $\omega$-stable and superstable theories.

1985-04-01

Louise Harrington , Michael Makkai

On considere la possibilite que la conjecture de Morley peut eventuellement etre demontree en donnant plus ou moins explicitement toutes les fonctions spectre possibles K(≥κ 1 )→I(T,K) avec chaque possibilite … On considere la possibilite que la conjecture de Morley peut eventuellement etre demontree en donnant plus ou moins explicitement toutes les fonctions spectre possibles K(≥κ 1 )→I(T,K) avec chaque possibilite se conformant a la condition de Morley