... 396 Notes and Papers Unimodular Minimal Structures Ehud Hrushovski, Ehud Hrushovski Department of Mathematics, MIT 2-277, Cambridge, ... IsraelSearch for more papers by this author Ehud Hrushovski, Ehud Hrushovski Department of Mathematics, MIT 2-277, Cambridge, MA ...
Tópico(s): Computability, Logic, AI Algorithms
1992 - Wiley | Journal of the London Mathematical Society
David M. Evans, Ehud Hrushovski,
... NR4 7TJSearch for more papers by this authorEhud Hrushovski, Ehud Hrushovski Department of Mathematics, Massachusetts Institute of Technology, ... NR4 7TJSearch for more papers by this authorEhud Hrushovski, Ehud Hrushovski Department of Mathematics, Massachusetts Institute of Technology, Cambridge, ...
Tópico(s): Polynomial and algebraic computation
1991 - Wiley | Proceedings of the London Mathematical Society
David M. Evans, Ehud Hrushovski,
... protected]Search for more papers by this authorEhud Hrushovski, Ehud Hrushovski Department of Mathematics, MIT, Cambridge, Massachusetts 02139, ... protected]Search for more papers by this authorEhud Hrushovski, Ehud Hrushovski Department of Mathematics, MIT, Cambridge, Massachusetts 02139, USA ...
Tópico(s): Coding theory and cryptography
1995 - Wiley | Journal of the London Mathematical Society
By Gregory Cherlin and Ehud Hrushovski: 193 pp., £35.00/£17.95, isbn 0-691-11331-9/0-691-11332-7(P) (Princeton University Press, 2003).
Tópico(s): Mathematics and Applications
2004 - Wiley | Bulletin of the London Mathematical Society
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Tópico(s): Finite Group Theory Research
2018 - Cambridge University Press | Bulletin of Symbolic Logic
Ehud Hrushovski, Silvain Rideau,
We introduce a class of theories called metastable, including the theory of algebraically closed valued fields ( $$\mathrm {ACVF}$$ ) as a motivating example. The key local notion is that of definable types dominated by their stable part. A theory is metastable (over a sort $$\Gamma $$ ) if every type over a sufficiently rich base structure can be viewed as part of a $$\Gamma $$ -parametrized family of stably dominated types. We initiate a study of definable groups in metastable theories of finite rank. Groups with a stably ...
Tópico(s): Algebraic Geometry and Number Theory
2019 - Birkhäuser | Selecta Mathematica
Ehud Hrushovski, Benjamin Martin, Silvain Rideau,
We prove that the theory of the p -adics {\mathbb Q}_p admits elimination of imaginaries provided we add a sort for {\mathrm GL}_n({\mathbb Q}_p)/{\mathrm GL}_n({\mathbb Z}_p) for each n . We also prove that the elimination of imaginaries is uniform in p . Using p -adic and motivic integration, we deduce the uniform rationality of certain formal zeta functions arising from definable equivalence relations. This also yields analogous results for definable equivalence relations over local fields of positive ...
Tópico(s): Algebraic Geometry and Number Theory
2018 - European Mathematical Society | Journal of the European Mathematical Society
Antoine Chambert-Loir, François Loeser,
... Poisson summation formula in motivic integration, established by Ehud Hrushovski and David Kazhdan (Moscow Math. J, 2009).
Tópico(s): Graph theory and applications
2016 - Johns Hopkins University Press | American Journal of Mathematics
Ehud Hrushovski, François Loeser,
We give a new proof - not using resolution of singularities - of a formula of Denef and the second author expressing the Lefschetz number of iterates of the monodromy of a function on a smooth complex algebraic variety in terms of the Euler characteristic of a space of truncated arcs. Our proof uses l-adic cohomology of non-archimedean spaces, motivic integration and the Lefschetz fixed point formula for finite order automorphisms. We also consider a generalization due to Nicaise and Sebag and at the ...
Tópico(s): Homotopy and Cohomology in Algebraic Topology
2015 - Société Mathématique de France | Annales Scientifiques de l École Normale Supérieure
Ehud Hrushovski, François Loeser, Bjorn Poonen,
Let K be a field that is complete with respect to a nonarchimedean absolute value such that K has a countable dense subset. We prove that the Berkovich analytification V^an of any d-dimensional quasi-projective scheme V over K embeds in R^{2d+1}. If, moreover, the value group of K is dense in R_{>0} and V is a curve, then we describe the homeomorphism type of V^an by using the theory of local dendrites.
Tópico(s): Advanced Algebra and Geometry
2015 - | L’Enseignement Mathématique
We attempt to formulate issues around modularity and Zilber's trichotomy in a setting that intersects additive combinatorics. In particular, we update the open problems on quasi-finite structures from [9].
Tópico(s): Graph theory and applications
2013 - Duke University Press | Notre Dame Journal of Formal Logic
Let T be a first-order theory. A correspondence is established between internal covers of models of T and definable groupoids within T. We also consider amalgamations of independent diagrams of algebraically closed substructures, and find strong relation between covers, uniqueness for 3-amalgamation, existence of 4-amalgamation, imaginaries of T^\si, and definable groupoids. As a corollary, we describe the imaginary elements of families of finite-dimensional vector spaces over pseudo-finite fields.
Tópico(s): Advanced Topics in Algebra
2012 - Scientific and Technological Research Council of Turkey (TUBITAK) | TURKISH JOURNAL OF MATHEMATICS
Let T be a first-order theory. A correspondence is established between internal covers of models of T and definable groupoids within T. We also consider amalgamations of independent diagrams of algebraically closed substructures, and find strong relation between covers, uniqueness for 3-amalgamation, existence of 4-amalgamation, imaginaries of T\si, and definable groupoids. As a corollary, we describe the imaginary elements of families of finite-dimensional vector spaces over pseudo-finite fields.
Tópico(s): Geometric and Algebraic Topology
2012 - Scientific and Technological Research Council of Turkey (TUBITAK) | TURKISH JOURNAL OF MATHEMATICS
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 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" ...
Tópico(s): Algebraic Geometry and Number Theory
2012 - American Mathematical Society | Transactions of the American Mathematical Society
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 G G , we show that a finite subset X X with | X X − 1 X | / | X | |X X ^{-1}X |/ |X| bounded is close to a finite subgroup, or else to a subset of a proper algebraic subgroup of G G . 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 ...
Tópico(s): Finite Group Theory Research
2011 - American Mathematical Society | Journal of the American Mathematical Society
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 (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 ...
Tópico(s): semigroups and automata theory
2011 - European Mathematical Society | Journal of the European Mathematical Society
Ehud Hrushovski, Ya’acov Peterzil, Anand Pillay,
We prove several structural results on definable, 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, (ii) ( 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 deducing the full compact domination conjecture for definably compact groups from the semisimple and commutative cases which were already ...
Tópico(s): Algebraic Geometry and Number Theory
2010 - Elsevier BV | Journal of Algebra
Ehud Hrushovski, David Kazhdan,
We develop a "motivic integration" version of the Poisson summation formula for function fields, with values in the Grothendieck ring of definable exponential sums.We also study division algebras over the function field, and obtain relations among the motivic Fourier transforms of a test function at different completions.We use these to prove, in a special case, a motivic version of a theorem of [DKV].
Tópico(s): Advanced Topics in Algebra
2009 - Independent University of Moscow | Moscow Mathematical Journal
Ehud Hrushovski, David Kazhdan, Nir Avni,
Tópico(s): Algebraic Geometry and Number Theory
2008 - Birkhäuser | Geometric and Functional Analysis
Zoé Chatzidakis, Ehud Hrushovski,
Abstract We draw a connection between the model-theoretic notions of modularity (or one-basedness), orthogonality and internality, as applied to difference fields, and questions of descent in in algebraic dynamics. In particular we prove in any dimension a strong dynamical version of Northcott's theorem for function fields, answering a question of Szpiro and Tucker and generalizing a theorem of Baker's for the projective line. The paper comes in three parts. This first part contains an exposition ...
Tópico(s): Homotopy and Cohomology in Algebraic Topology
2008 - Cambridge University Press | Journal of the Institute of Mathematics of Jussieu
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 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 ...
Tópico(s): Homotopy and Cohomology in Algebraic Topology
2007 - American Mathematical Society | Journal of the American Mathematical Society
Ehud Hrushovski, Ya’acov Peterzil,
Abstract We use a new construction of an o-minimal structure, due to Lipshitz and Robinson, to answer a question of van den Dries regarding the relationship between arbitrary o-minimal expansions of real closed fields and structures over the real numbers. We write a first order sentence which is true in the Lipshitz-Robinson structure but fails in any possible interpretation over the field of real numbers.
Tópico(s): Computability, Logic, AI Algorithms
2007 - Cambridge University Press | Journal of Symbolic Logic
Deirdre Haskell, Ehud Hrushovski, Dugald Macpherson,
It is shown that if K is an algebraically closed valued field with valuation ring R, then Th(K) has elimination of imaginaries if sorts are added whose elements are certain cosets in Kn of certain definable R-submodules of Kn (for all ). The proof involves the development of a theory of independence for unary types, which play the role of 1-types, followed by an analysis of germs of definable functions from unary sets to the sorts.
Tópico(s): Mathematical and Theoretical Analysis
2006 - De Gruyter | Journal für die reine und angewandte Mathematik (Crelles Journal)
Ehud Hrushovski, Itamar Pitowsky,
Tópico(s): Quantum Information and Cryptography
2004 - Elsevier BV | Studies in History and Philosophy of Science Part B Studies in History and Philosophy of Modern Physics
Ehud Hrushovski, Masanori Itai,
We develop a geometric approach to definable sets in differentially closed fields, with emphasis on the question of orthogonality to a given strongly minimal set. Equivalently, within a family of ordinary differential equations, we consider those equations that can be transformed, by differential-algebraic transformations, so as to yield solutions of a given fixed first-order ODE X X . We show that this sub-family is usually definable (in particular if X X lives on a curve of positive genus). As ...
Tópico(s): Advanced Numerical Analysis Techniques
2003 - American Mathematical Society | Transactions of the American Mathematical Society
Zoé Chatzidakis, Ehud Hrushovski, Ya’acov Peterzil,
We classify all possible combinatorial geometries associated with one-dimensional difference equations, in any characteristic. The theory of difference fields admits a proper interpretation of itself, namely the reduct replacing the automorphism by its nth power. We show that these reducts admit a successively smoother theory as n becomes large; and we succeed in defining a limit structure to these reducts, or rather to the structure they induce on one-dimensional sets. This limit structure is shown ...
Tópico(s): Polynomial and algebraic computation
2002 - Wiley | Proceedings of the London Mathematical Society
Using methods of geometric stability (sometimes generalized to finite S1 rank), we determine the structure of Abelian groups definable in ACFA, the model companion of fields with an automorphism. We also give general bounds on sets definable in ACFA. We show that these tools can be used to study torsion points on Abelian varieties; among other results, we deduce a fairly general case of a conjecture of Tate and Voloch on p-adic distances of torsion points from subvarieties.
Tópico(s): Geometric and Algebraic Topology
2001 - Elsevier BV | Annals of Pure and Applied Logic
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 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 ...
Tópico(s): Analytic Number Theory Research
2000 - Johns Hopkins University Press | American Journal of Mathematics
Bradd Hart, Ehud Hrushovski, M. Laskowski,
Let T be a complete, first-order theory in a finite or countable language having infinite models.Let I(T, κ) be the number of isomorphism types of models of T of cardinality κ.We denote by µ (respectively μ) the number of cardinals (respectively infinite cardinals) less than or equal to κ.Theorem.I(T, κ), as a function of κ > ℵ 0 , is the minimum of 2 κ and one of the following functions:4. the constant function 2 ;5. d+1 (µ) for some infinite, countable ordinal d;6. d i=1 Γ(i) where d is an integer greater ...
Tópico(s): Logic, Reasoning, and Knowledge
2000 - Princeton University | Annals of Mathematics
Zoé Chatzidakis, Ehud Hrushovski,
A difference field is a field with a distinguished automorphism σ \sigma . 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 0 0 .
Tópico(s): Algebraic Geometry and Number Theory
1999 - American Mathematical Society | Transactions of the American Mathematical Society