Measured creatures

We prove that two basic questions on outer measure are undecidable. First we show that consistently every sup-measurable functionf: ℝ² → ℝ is measurable. The interest in sup-measurable functions comes from differential equations and the question for which functionsf: ℝ² → ℝ the Cauchy problemy′=f(x,y), y(x₀)=y₀ has a unique almost-everywhere solution in the classAC _t(ℝ) of locally absolutely continuous functions on ℝ. Next we prove that consistently every functionf: ℝ → ℝ is continuous on some set of positive outer Lebesgue measure. This says that in a strong sense the family of continuous functions (from the reals to the reals) is dense in the space of arbitrary such functions. For the proofs we discover and investigate a new family of nicely definable forcing notions (so indirectly we deal with nice ideals of subsets of the reals—the two classical ones being the ideal of null sets and the ideal of meagre ones). Concerning the method, i.e., the development of a family of forcing notions, the point is that whereas there are many such objects close to the Cohen forcing (corresponding to the ideal of meagre sets), little has been known on the existence of relatives of the random real forcing (corresponding to the ideal of null sets), and we look exactly at such forcing notions. The first author thanks The Hebrew University of Jerusalem for support during his visits to Jerusalem and the KBN (Polish Committee of Scientific Research) for partial support through grant 2P03A03114. The research of the second author was partially supported by the Israel Science Foundation. Publication 736.     

Hechler’s theorem for the null ideal

We prove the following theorem: For a partially ordered set Q such that every countable subset of Q has a strict upper bound, there is a forcing notion satisfying the countable chain condition such that, in the forcing extension, there is a basis of the null ideal of the real line which is order-isomorphic to Q with respect to set-inclusion. This is a variation of Hechlers classical result in the theory of forcing. The corresponding theorem for the meager ideal was established by Bartoszyski and Kada.Research supported by NSERC. The first author thanks F.D. Tall and the Department of Mathematics at the University of Toronto for their hospitality during the academic year 2003/2004 when the present paper was completed.The second author was supported by Grant-in-Aid for Young Scientists (B) 14740058, MEXT.Mathematics Subject Classification (2000): 03E35, 03E17Revised version: 16 February 2004     

Specialising Aronszajn trees by countable approximations

We show that there are proper forcings based upon countable trees of creatures that specialise a given Aronszajn tree.The first author was partially supported by a Minerva fellowship. The second author's research was partially supported by the ``Israel Science Foundation', founded by the Israel Academy of Science and Humanities. This is the second author's work number 778. Mathematics Subject Classification (2000): 03E15, 03E17, 03E35, 03D65     

Proper forcing extensions and Solovay models

We study the preservation of the property of being a Solovay model under proper projective forcing extensions. We show that every strongly-proper forcing notion preserves this property. This yields that the consistency strength of the absoluteness of under strongly-proper forcing notions is that of the existence of an inaccessible cardinal. Further, the absoluteness of under projective strongly-proper forcing notions is consistent relative to the existence of a -Mahlo cardinal. We also show that the consistency strength of the absoluteness of under forcing extensions with -linked forcing notions is exactly that of the existence of a Mahlo cardinal, in contrast with the general ccc case, which requires a weakly-compact cardinal.Research partially supported by the research projects BFM2002-03236 of the Spanish Ministry of Science and Technology, and 2002SGR 00126 of the Generalitat de Catalunya. The second author was also partially supported by the research project GE01/HUM10, Grupos de excelencia, Principado de Asturias.Mathematics Subject Classification (2000): 03E15, 03E35     

Adjoining cofinitary permutations (II)

 We construct several forcing models in each of which there exists a maximal cofinitary group, i.e., a maximal almost disjoint group, G≤Sym(ℕ), such that G is also a maximal almost disjoint family in Sym(ℕ). We also ask several open questions in this area in the fourth section of this paper. Received: 25 December 2000 / Revised version: 10 December 2001 Published online: 5 November 2002 Current address:Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109-1109. USA.e-mail: yizhang@umich.edu The author's research on this subject was partially supported by a visiting grant from the Institute Mittag–Leffler, Royal Academy of Science, Sweden and the grant no. 40734 of Academy of Finland. Mathematics Subject Classification (2000): 03E35, 20A15, 20B07, 20B35     

Strongly meager sets of size continuum

We will construct several models where there are no strongly meager sets of size 2⁰. First author partially supported by NSF grant DMS 0200671.Second author partially supported by Israel Science Foundation and NSF grant DMS 0072560. Publication 807. Mathematics Subject Classification (2000):03E15, 03E20     

Gently killing S-spaces

We produce a model of ZFC in which there are no locally compact first countable S-spaces, and in which 2^ℵ ₀<2^ℵ ₁. A consequence of this is that in this model there are no locally compact, separable, hereditarily normal spaces of size ℵ₁, answering a question of the second author [9]. The first author would like to thank The Hebrew University of Jerusalem for their support while the research in this paper was being carried out. The research of the second author was partially supported by NSF Grant DMS-9322613. The research of the third author was partially supported by NSF grant DMS-9704477 and the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities. This is publication number 690 in the list of the third author.     

Assouad-Nagata dimension via Lipschitz extensions

In the first part of the paper we show how to relate several dimension theories (asymptotic dimension with Higson property, asymptotic dimension of Gromov and capacity dimension of Buyalo [7]) to Assouad-Nagata dimension. This is done by applying two functors on the Lipschitz category of metric spaces: microscopic and macroscopic. In the second part we identify (among spaces of finite Assouad-Nagata dimension) spaces of Assouad-Nagata dimension at most n as those for which the n-sphere S ⁿ is a Lipschitz extensor. Large scale and small scale analogues of that result are given. The author was partially supported by Grant No.2004047 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel. The author was supported by Grant AP2004-2494 from the Ministerio de Educacion y Ciencia, Spain. He thanks the Department of Mathematics of University of Tennessee for their hospitality.     

Initial segments of the degrees of size ℵ1

We settle a series of questions first raised by Yates at the Jerusalem (1968) Colloquium on Mathematical Logic by characterizing the initial segments of the degrees of unsolvability of size ℵ₁: Every upper semi-lattice of size ℵ₁ with zero, in which every element has at most countably many predecessors, is isomorphic to an initial segment of the Turing degrees. The second author was partially supported by a grant from the NSF. The research was carried out while he was on sabbatical leave from Cornell University and a Visiting Professor at the Hebrew University, Jersalem. He would like to thank the Hebrew University and in particular the logicians there for their hospitality.     

Strongly meager and strong measure zero sets

 In this paper we present two consistency results concerning the existence of large strong measure zero and strongly meager sets. RID="ID=" <E5>Mathematics Subject Classification (2000):</E5>&ensp;03E35 RID="ID=" The first author was supported by Alexander von Humboldt Foundation and NSF grant DMS 95-05375. The second author was partially supported by Basic Research Fund, Israel Academy of Sciences, publication 658 Received: 6 January 1999 / Revised version: 20 July 1999 / Published online: 25 February 2002 RID=" ID=" <E5>Mathematics Subject Classification (2000):</E5>&ensp;03E35 RID=" ID=" The first author was supported by Alexander von Humboldt Foundation and NSF grant DMS 95-05375. The second author was partially supported by Basic Research Fund, Israel Academy of Sciences, publication 658     

A construction of all normal subgroup lattices of 2-transitive automorphism groups of linearly ordered sets

We give a complete classification and construction of all normal subgroup lattices of 2-transitive automorphism groupsA(Ω) of linearly ordered sets (Ω, ≦). We also show that in each of these normal subgroup lattices, the partially ordered subset of all those elements which are finitely generated as normal subgroups forms a lattice which is closed under even countably-infinite intersections, and we derive several further group-theoretical consequences from our classification. This research was supported by an award from the Minerva-Stiftung, München. The work was done during a stay of the first-named author at The Hebrew University of Jerusalem in fall 1982. He would like to thank his colleagues in Jerusalem for their hospitality and a wonderful time.     

On potential isomorphism and non-structure

We show in the paper that for any non-classifiable countable theory T there are non-isomorphic models and that can be forced to be isomorphic without adding subsets of small cardinality. By making suitable cardinal arithmetic assumptions we can often preserve stationary sets as well. We also study non-structure theorems relative to the Ehrenfeucht-Fraïssé game. The research of the first and second author was partially supported by Academy of Finland grant 40734 Mathematics Subject Classification (2000): 03C55, 03C45     

Representations of symplectic reflection algebras and resolutions of deformations of symplectic quotient singularities

We give an equivalence of triangulated categories between the derived category of finitely generated representations of symplectic reflection algebras associated with wreath products (with parameter t=0) and the derived category of coherent sheaves on a crepant resolution of the spectrum of the centre of these algebras.Mathematics Subject Classification (2000): 14E15, 16Rxx, 16S38, 18E30The first author was partially supported by the Nuffield Foundation grant NAL/00625/G and by the University of Washingtons Milliman Fund. The second author was partially supported by an NSF grant DMS-0070560. Both authors are grateful to the Edinburgh Mathematical Society and the Leverhulme Trust for support.Dedicated to Claus Michael Ringel on the occasion of his sixtieth birthday     

Motives of some Fano varieties

We study the Fano varieties of projective k-planes lying in hypersurfaces and investigate the associated motives. The first author is partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada. The second author is partially supported by TüBİTAK-BDP funds and Bilkent University research development funds.     

Consistent parameter estimation for partially observed diffusions with small noise

In this paper we provide a consistency result for the MLE for partially observed diffusion processes with small noise intensities. We prove that if the underlying deterministic system enjoys an identifiability property, then any MLE is close to the true parameter if the noise intensities are small enough. The proof uses large deviations limits obtained by PDE vanishing viscosity methods. A deterministic method of parameter estimation is formulated. We also specialize our results to a binary detection problem, and compare deterministic and stochastic notions of identifiability.This research was supported: by Systems Research Center, University of Maryland through NSF Grant CDR-85-00108 and AFOSR-URI Grant 87-0073; by Lefschetz Center for Dynamical Systems, Division of Applied Mathematics, Brown University, under ARO/MIT Grant DAAL-03-86-K-0171; by INRIA Sophia Antipolis, under ERO/INRIA Grant DAJA45-90-C-0008, and by the CNRS-GRAutomatique.     

Large deviations and stochastic flows of diffeomorphisms

Summary Previous results in the theory of large deviations for additive functionals of a diffusion process on a compact manifold M are extended and then applied to the analysis of the Lyapunov exponents of a stochastic flow of diffeomorphisms of M. An approximation argument relates these results to the behavior near the diagonal Δ in M ² of the associated two point motion. Finally it is shown, under appropriate non-degeneracy conditions, that the two-point motion is ergodic on M ²-Δ if the top Lyapunov exponent is positive. At the period when this research was initiated, both authors where guests of the I.M.A. in Minneapolis. The first author was at Aberdeen University, Scotland when this article was prepared. Throughout the period of this research, the second author has been partially supported by N.S.F. grant DMS-8611487 and ARO grant DAAL03-86-K-171     

Dirac Cohomology of some Harish-Chandra Modules

In this paper we determine the Dirac cohomology of certain irreducible Harish-Chandra modules of a semisimple connected Lie group G with finite center: irreducible finite-dimensional modules and unitary A (λ) modules. We also comment on the relationship to (, K)-cohomology. The first named author is partially supported by research grants from RGC of Hong Kong SAR and NSF of China. The second named author is partially supported by the National Natural Science Foundation of China (10501025), Liu Hui Center for Applied Mathematics and Youth Teachers Foundation of Tianjin University. The third named author is partially supported by a grant from the Ministry of Science, Education and Sports of the Republic of Croatia.     

On nice equivalence relations on <Superscript>λ</Superscript>2

Let E be an equivalence relation on the powerset of an uncountable set, which is reasonably definable. We assume that any two subsets with symmetric difference of size exactly 1 are not equivalent. We investigate whether for E there are many pairwise non equivalent sets. I would like to thank Alice Leonhardt for the beautiful typing.This research was supported by The Israel Science Foundation. Publication 724. Mathematics Subject Classification (2000): 03E47, 03E35; 20K20, 20K35     

Pluricanonical systems on irregular 3-folds of general type

In this paper we prove that if X is an irregular 3-fold with χ(ω _X ) > 0, then |mK _X | is birational for all m ≥ 5.The first author was partially supported by NSC and NCTS of Taiwan. The second author was partially supported by NSF research grant no: 0456363.     

The unitary representation theory of GL(n) of an infinite discrete field

IfK is an infinite field and ifG=GL(n, K) with the discrete topology, then all principal-series representations ofG are irreducible, and any two such with the same central character ψ are weakly equivalent to one another and to the ψ-regular representation. In addition, every irreducible unitary representation ofG which is not one-dimensional weakly contains a representation of the principal series. We deduce that every maximal ideal ofC*(G) is either of codimension 1 or else a kernel of a principal-series representation. In particular, except in the exceptional case whereK is an infinite algebraic extension of a finite field, the reducedC*-algebra of PGL(n, K) is simple, as was also shown by de la Harpe in many cases. Partially supported by NSF Grant DMS-85-06130. It is a pleasure also to acknowledge the hospitality of the Institute for Advanced Studies, The Hebrew University of Jerusalem, 91904 Jerusalem, Israel, from January to August, 1988. Partially supported by NSF Grants DMS-84-00900 and DMS-87-00551. Much of this work was done while visiting at, and partially supported by, the Department of Mathematics and Computer Science, Bar-Ilan University, 52100 Ramat Gan, Israel.