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Anand Pillay [75]A. Pillay [17]Ap Pillay [1]
  1. Jakub Gismatullin, Davide Penazzi & Anand Pillay (2014). On Compactifications and the Topological Dynamics of Definable Groups. Annals of Pure and Applied Logic 165 (2):552-562.
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  2. Krzysztof Krupiński, Anand Pillay & Sławomir Solecki (2013). Borel Equivalence Relations and Lascar Strong Types. Journal of Mathematical Logic 13 (2):1350008.
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  3. Anand Pillay (2013). Topological Dynamics and Definable Groups. Journal of Symbolic Logic 78 (2):657-666.
    We give a commentary on Newelski's suggestion or conjecture [8] that topological dynamics, in the sense of Ellis [3], applied to the action of a definable group $G(M)$ on its “external type space” $S_{G,\textit{ext}}(M)$, can explain, account for, or give rise to, the quotient $G/G^{00}$, at least for suitable groups in NIP theories. We give a positive answer for measure-stable (or $fsg$) groups in NIP theories. As part of our analysis we show the existence of “externally definable” generics of $G(M)$ (...)
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  4. Anand Pillay (2013). Weight and Measure in NIP Theories. Notre Dame Journal of Formal Logic 54 (3-4):567-578.
    We initiate an account of Shelah’s notion of “strong dependence” in terms of generically stable measures, proving a measure analogue of the fact that a stable theory $T$ is “strongly dependent” if and only if all types have almost finite weight.
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  5. Ehud Hrushovski, Anand Pillay & Pierre Simon (2012). A Note on Generically Stable Measures and Fsg Groups. Notre Dame Journal of Formal Logic 53 (4):599-605.
    We prove (Proposition 2.1) that if $\mu$ is a generically stable measure in an NIP (no independence property) theory, and $\mu(\phi(x,b))=0$ for all $b$ , then for some $n$ , $\mu^{(n)}(\exists y(\phi(x_{1},y)\wedge \cdots \wedge\phi(x_{n},y)))=0$ . As a consequence we show (Proposition 3.2) that if $G$ is a definable group with fsg (finitely satisfiable generics) in an NIP theory, and $X$ is a definable subset of $G$ , then $X$ is generic if and only if every translate of $X$ does not (...)
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  6. Charlotte Kestner & Anand Pillay (2011). Remarks on Unimodularity. Journal of Symbolic Logic 76 (4):1453-1458.
    We clarify and correct some statements and results in the literature concerning unimodularity in the sense of Hrushovski [7], and measurability in the sense of Macpherson and Steinhorn [8], pointing out in particular that the two notions coincide for strongly minimal structures and that another property from [7] is strictly weaker, as well as "completing" Elwes' proof [5] that measurability implies 1-basedness for stable theories.
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  7. Anand Pillay (2011). Stable Embeddedness and Nip. Journal of Symbolic Logic 76 (2):665 - 672.
    We give some sufficient conditions for a predicate P in a complete theory T to be "stably embedded". Let P be P with its "induced θ-definable structure". The conditions are that P (or rather its theory) is "rosy", P has NIP in T and that P is stably 1-embedded in T. This generalizes a recent result of Hasson and Onshuus [6] which deals with the case where P is o-minimal in T. Our proofs make use of the theory of strict (...)
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  8. Patricia Blanchette, Kit Fine, Heike Mildenberger, André Nies, Anand Pillay, Alexander Razborov, Alexandra Shlapentokh, John R. Steel & Boris Zilber (2009). Notre Dame, Indiana May 20–May 23, 2009. Bulletin of Symbolic Logic 15 (4).
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  9. Anand Pillay & Wai Yan Pong (2009). Corrigendum To: "On Lascar Rank and Morley Rank of Definable Groups in Differentially Closed Fields". Journal of Symbolic Logic 74 (4):1436 - 1437.
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  10. S. Barry Cooper, Herman Geuvers, Anand Pillay & Jouko Väänänen (2008). Preface. Annals of Pure and Applied Logic 156 (1):1-2.
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  11. Clifton Ealy, Krzysztof Krupiński & Anand Pillay (2008). Superrosy Dependent Groups Having Finitely Satisfiable Generics. Annals of Pure and Applied Logic 151 (1):1-21.
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  12. Anand Pillay (2007). Imaginaries in Pairs of Algebraically Closed Fields. Annals of Pure and Applied Logic 146 (1):13-20.
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  13. Anand Pillay & Dominika Polkowska (2006). On PAC and Bounded Substructures of a Stable Structure. Journal of Symbolic Logic 71 (2):460 - 472.
    We introduce and study the notions of a PAC-substructure of a stable structure, and a bounded substructure of an arbitrary substructure, generalizing [10]. We give precise definitions and equivalences, saying what it means for properties such as PAC to be first order, study some examples (such as differentially closed fields) in detail, relate the material to generic automorphisms, and generalize a "descent theorem" for pseudo-algebraically closed fields to the stable context. We also point out that the elementary invariants of pseudo-algebraically (...)
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  14. Alessandro Berarducci, Margarita Otero, Yaa’cov Peterzil & Anand Pillay (2005). A Descending Chain Condition for Groups Definable in o-Minimal Structures. Annals of Pure and Applied Logic 134 (2):303-313.
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  15. Anand Pillay & Evgueni Vassiliev (2005). On Lovely Pairs and the (∃ y ∈ P ) Quantifier. Notre Dame Journal of Formal Logic 46 (4):491-501.
    Given a lovely pair P ≺ M of models of a simple theory T, we study the structure whose universe is P and whose relations are the traces on P of definable (in ℒ with parameters from M) sets in M. We give a necessary and sufficient condition on T (which we call weak lowness) for this structure to have quantifier-elimination. We give an example of a non-weakly-low simple theory.
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  16. Anand Pillay (2004). Type-Definability, Compact Lie Groups, and o-Minimality. Journal of Mathematical Logic 4 (02):147-162.
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  17. Anand Pillay & Martin Ziegler (2004). On a Question of Herzog and Rothmaler. Journal of Symbolic Logic 69 (2):478-481.
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  18. Itay Ben-Yaacov, Anand Pillay & Evgueni Vassiliev (2003). Lovely Pairs of Models. Annals of Pure and Applied Logic 122 (1-3):235-261.
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  19. Anand Pillay (2003). On Countable Simple Unidimensional Theories. Journal of Symbolic Logic 68 (4):1377-1384.
    We prove that any countable simple unidimensional theory T is supersimple, under the additional assumptions that T eliminates hyperimaginaries and that the $D_\phi-ranks$ are finite and definable.
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  20. Anand Pillay & Wai Yan Pong (2002). On Lascar Rank and Morley Rank of Definable Groups in Differentially Closed Fields. Journal of Symbolic Logic 67 (3):1189-1196.
    Morley rank and Lascar rank are equal on generic types of definable groups in differentially closed fields with finitely many commuting derivations.
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  21. Anand Pillay & Thomas Scanlon (2002). Compact Complex Manifolds with the DOP and Other Properties. Journal of Symbolic Logic 67 (2):737-743.
    We point out that a certain complex compact manifold constructed by Lieberman has the dimensional order property, and has U-rank different from Morley rank. We also give a sufficient condition for a Kahler manifold to be totally degenerate (that is, to be an indiscernible set, in its canonical language) and point out that there are K3 surfaces which satisfy these conditions.
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  22. Anand Pillay & Mark D. Schlatter (2002). Some Results on Permutation Group Isomorphism and Categoricity. Journal of Symbolic Logic 67 (3):910-914.
    We extend Morley's Theorem to show that if a theory is κ-p-categorical for some uncountable cardinal κ, it is uncountably categorical. We then discuss ω-p-categoricity and provide examples to show that similar extensions for the Baldwin-Lachlan and Lachlan Theorems are not possible.
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  23. Samuel R. Buss, Alexander S. Kechris, Anand Pillay & Richard A. Shore (2001). The Prospects for Mathematical Logic in the Twenty-First Century. Bulletin of Symbolic Logic 7 (2):169-196.
    The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.
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  24. E. Casanovas, D. Lascar, A. Pillay & M. Ziegler (2001). Galois Groups of First Order Theories. Journal of Mathematical Logic 1 (02):305-319.
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  25. D. Lascar & A. Pillay (2001). Hyperimaginaries and Automorphism Groups. Journal of Symbolic Logic 66 (1):127-143.
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  26. Anand Pillay (2001). A Note on Existentially Closed Difference Fields with Algebraically Closed Fixed Field. Journal of Symbolic Logic 66 (2):719-721.
    We point out that the theory of difference fields with algebraically closed fixed field has no model companion.
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  27. Andreas Baudisch & Anand Pillay (2000). A Free Pseudospace. Journal of Symbolic Logic 65 (1):443-460.
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  28. Bradd Hart, Byunghan Kim & Anand Pillay (2000). Coordinatisation and Canonical Bases in Simple Theories. Journal of Symbolic Logic 65 (1):293-309.
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  29. Hirotaka Kikyo & Anand Pillay (2000). The Definable Multiplicity Property and Generic Automorphisms. Annals of Pure and Applied Logic 106 (1-3):263-273.
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  30. A. Pillay, D. Hallett, G. Hjorth, C. Jockusch, A. Kanamori, H. J. Keisler & V. McGee (2000). 2000 Annual Meeting of the Association for Symbolic Logic. Bulletin of Symbolic Logic 6 (3):361-396.
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  31. A. Pillay, D. Hallett, G. Hjorth, C. Jockusch, A. Kanamori, H. J. Keisler & V. McGee (2000). University of Illinois at Urbana-Champaign, June 3–7, 2000. Bulletin of Symbolic Logic 6 (3).
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  32. Anand Pillay (2000). A Note on CM-Triviality and the Geometry of Forking. Journal of Symbolic Logic 65 (1):474-480.
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  33. Daniel Lascar & Anand Pillay (1999). Forking and Fundamental Order in Simple Theories. Journal of Symbolic Logic 64 (3):1155-1158.
    We give a characterisation of forking in the context of simple theories in terms of the fundamental order.
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  34. A. Pillay (1999). Review: Ehud Hrushovski, Boris Zilber, Zariski Geometries. [REVIEW] Journal of Symbolic Logic 64 (2):906-908.
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  35. T. Arai, Z. Chatzidakis & A. Pillay (1998). Laforte, G., See Downey, R. Annals of Pure and Applied Logic 95:287.
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  36. Z. Chatzidakis & A. Pillay (1998). Generic Structures and Simple Theories. Annals of Pure and Applied Logic 95 (1-3):71-92.
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  37. Koichiro Ikeda, Akito Tsuboi & Anand Pillay (1998). On Theories Having Three Countable Models. Mathematical Logic Quarterly 44 (2):161-166.
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  38. Byunghan Kim & Anand Pillay (1998). From Stability to Simplicity. Bulletin of Symbolic Logic 4 (1):17-36.
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  39. Anand Pillay (1998). Definability and Definable Groups in Simple Theories. Journal of Symbolic Logic 63 (3):788-796.
    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.
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  40. Byunghan Kim & Anand Pillay (1997). Simple Theories. Annals of Pure and Applied Logic 88 (2):149-164.
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  41. Anand Pillay (1997). Differential Galois Theory II. Annals of Pure and Applied Logic 88 (2-3):181-191.
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  42. Anand Pillay (1997). Remarks on Galois Cohomology and Definability. Journal of Symbolic Logic 62 (2):487-492.
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  43. Anand Pillay & Akito Tsuboi (1997). Amalgamations Preserving ℵ0-Categoricity. Journal of Symbolic Logic 62 (4):1070 - 1074.
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  44. Anand Pillay (1996). Definable Sets in Generic Complex Tori. Annals of Pure and Applied Logic 77 (1):75-80.
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  45. B. Hart, A. Pillay & S. Starchenko (1995). 1-Based Theories — the Main Gap for a -Models. Archive for Mathematical Logic 34 (5):285-300.
    We prove the Main Gap for the class of a -models (sufficiently saturated models) of an arbitrary stable 1-based theory T . We (i) prove a strong structure theorem for a -models, assuming NDOP, and (ii) roughly compute the number of a -models of T in any given cardinality. The analysis uses heavily group existence theorems in 1-based theories.
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  46. Anand Pillay (1995). Review: Wilfrid Hodges, Model Theory. [REVIEW] Journal of Symbolic Logic 60 (2):689-691.
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  47. Anand Pillay (1995). The Geometry of Forking and Groups of Finite Morley Rank. Journal of Symbolic Logic 60 (4):1251-1259.
    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.
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  48. Anand Pillay & Bruno Poizat (1995). Bodies and Surgery. Journal of Symbolic Logic 60 (2):528-533.
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  49. Anand Pillay & Bruno Poizat (1995). Corps Et Chirurgie. Journal of Symbolic Logic 60 (2):528-533.
    Les corps algébriquement clos, réels clos et pseudo-finis n'ont, pour chaque entier n, qu'un nombre fini d'extensions de degré n; nous montrons qu'ils partagent cette propriété avec tous les corps qui, comme eux, satisfont une propriété très rudimentaire de préservation de la dimension, de nature modèle-théorique. Ce résultat est atteint en montrant qu'une certaine action du groupe GLn d'un tel corps n'a qu'un nombre fini d'orbites. /// La korpoj algebre fermataj, reale fermataj kaj pseudofinataj ne havas, pri ciu integro n, (...)
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  50. Ambar Chowdhury & Anand Pillay (1994). On the Number of Models of Uncountable Theories. Journal of Symbolic Logic 59 (4):1285-1300.
    In this paper we establish the following theorems. THEOREM A. Let T be a complete first-order theory which is uncountable. Then: (i) I(|T|, T) ≥ ℵ 0 . (ii) If T is not unidimensional, then for any λ ≥ |T|, I (λ, T) ≥ ℵ 0 . THEOREM B. Let T be superstable, not totally transcendental and nonmultidimensional. Let θ(x) be a formula of least R ∞ rank which does not have Morley rank, and let p be any stationary completion (...)
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