arXiv subjects
Publications and source records attributed to Anand Pillay.
We settle some open problems in the special case of groups in o-minimal structures, such as the equality of G^00 and G^000 and the equivalence of definable amenability and existence of a type with bounded orbit. We prove almost exactness of the G to G^00 functor. We ask further questions about types with bounded orbits in NIP theories.
We clarify the relationship between unimodulariy in the sense of Hrushovski and measurability in the sense of Macpherson and Steinhorn, correcting some statements in the literature. In particular we point out that the notions coincide for strongly minimal sets.
We study the notions generic stability, regularity, homogeneous pregeometries, quasiminimality, and their mutual relations, in an arbitrary first order theory T. We prove that "infinite-dimensional homogeneous pregeometries" coincide with generically stable strongly regular types (p(x),x=x). We prove that quasiminimal structures of cardinality at least aleph-2 are homogeneous pregeometries, We prove that the generic type of an arbitrary quasiminimal structure is locally strongly regular. Some of the results depend on a general dichotomy for regular-like types: generic stability, or existence of a suitable definable partial ordering.
We study the model theory of covers of groups definable in o-minimal structures. This includes the case of covers of compact real Lie groups. In particular we study categoricity questions, pointing out some notable differences with the case of covers of complex algebraic groups studied by Zilber and his students. We also discuss from a model-theoretic point of view the following question, related to "Milnor's conjecture": is a finite central extension (as an abstract group) of a compact Lie group also a topological extension?
Essential $\aleph_0$-categoricity; i.e., $\aleph_0$-categoricity in some full countable language, is shown to be a robust notion for strongly minimal compact complex manifolds. Characterisations of triviality and essential $\aleph_0$-categoricity are given in terms of complex-analytic automorphisms, in the simply connected case, and correspondences in general. As a consequence it is pointed out that an example of McMullen yields a strongly minimal compact Kähler manifold with trivial geometry but which is not $\aleph_0$-categorical, giving a counterexample to a conjecture of the second author and Tom Scanlon.
I give an account of Shelah's notion of strong dependence, or strong NIP, in terms of suitable generically stable measures, forking, and weight.
We study stable like behaviour in first order theories without the independence property. We introduce generically stable measures, give characterizatiions, and show their ubiquity. We also introduce generic compact domination. We also prove the approximate definability of arbitrary Borel probability measures on definable sets in the real and p-adic fields.
We give sufficient conditions for a predicate P in a complete theory T to be stably embedded: P with its induced 0-definable structure has "finite rank", P has NIP in T and P is 1-stably embedded. This generalizes recent work by Hasson and Onshuus in the case where P is o-minimal in T.
We prove that if K is an infinite stable field whose generic type has weight 1 then K is separably closed. We also obtain partial results when the generic of K has finite weight and when K is strongly stable.
We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of $NIP$ (not the independence property), continuing aspects of math.LO/0607442. Among key results are: (i) if $p = tp(b/A)$ 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 $bdd(A)$ (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 translation invariant Keisler measures on groups with finitely satisfiable generics (vi) A proof of the compact domination conjecture for definably compact commutative groups in $o$-minimal expansions of real closed fields.
I prove that the generic type of the free nonabelian group has infinite weight (strengthening non superstability of the free group).
We prove an analogue of the Lindemann-Weierstrass theorem (that the exponentials of Q-linearly independent algebraic numbers are algebraically independent) for commutative algebraic groups G without unipotent quotients, over function fields. We concentrate on solutions to the the differential algebraic relations satisfied by exp from LG to G.
We prove several structural results on definably compact groups G in o-minimal expansions of real closed fields, such as (i) G is definably an almost direct product of a semisimple group and a commutative group, and (ii) the group (G, .) is elementarily equivalent to (G/G^00, .). We also prove results on the internality of finite covers of G in an o-minimal environment, as well as the full compact domination conjecture. These results depend on key theorems about the interpretability of central and finite extensions of definable groups, in the o-minimal context. These methods and others also yield interpretability results for universal covers of arbitrary definable real Lie groups, from which we can deduce the semialgebraicity of finite covers of Lie groups such as SL(2,R).
We study a model theoretic context (finite thorn rank, NIP, with finitely satisfiable generics) which is a common generalization of groups of finite Morley rank and definably compact groups in o-minimal structures. We show that assuming thorn rank 1, the group is abelian-by-finite, and assuming thorn rank 2 the group is solvable by finite. Also a field is algebraically closed.
We survey and explain some recent work at the intersection of model theory and bimeromorphic geometry (classification of compact complex manifolds). Included here are the essential saturation of the many sorted structure $\mathcal{C}$ of compact Kaehler manifolds, the conjectural role of hyperkaehler manifolds in the description of strongly minimal sets in $\mathcal{C}$ and Campana's work on the isotriviality of hyperkaehler families and its connection with the nonmultidimensionality conjecture.
We study model-theoretic and stability-theoretic properies of the nonabelian free group in the light of Sela's recent result on stability and results announced by Bestvina and Feighn. We point out analogies between the free group and so-called bad groups of finite Morley rank and we prove non CM-triviality of the free group.
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 a new notion "compact domination" (domination of a definable set by a compact space) and raise some new conjectures in the $o$-minimal case.
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.