Complete Hartogs domains in C2 have regular Bergman and Szegő projections

The first author partially supported by NSF grant DMS-8701038     

A partition relation for successors of Large Cardinals

We address partition problems of Erd?s and Hajnal by showing that for all , if and carries a -dense ideal. If is measurable we show that for where is a very large ordinal less than that is closed under all primitive recursive ordinal operations. Received: 27 June 2001 / Revised version: 5 December 2001 / Published online: 4 February 2003 The first author was partially supported by NSF grant DMS-0101155 and the Equipe d'Analyse Univ. of Paris 6. The second author was partially supported by NSF grants DMS-0072560 and DMS-9704477.     

Sharp trace theorems for null hypersurfaces on Einstein metrics with finite curvature flux

The main objective of the paper is to prove a geometric version of sharp trace and product estimates on null hypersurfaces with finite curvature flux. These estimates play a crucial role to control the geometry of such null hypersurfaces. The paper is based on an invariant version of the classical Littlewood–Paley theory, in a noncommutative setting, defined via heat flow on surfaces. Received: April 2004 Revision: June 2005 Accepted: July 2005 The first author is partially supported by NSF grant DMS-0070696. The second author is a Clay Prize Fellow and is partially supported by NSF grant DMS-01007791.     

Classification of manifolds with weakly 1/4-pinched curvatures

We show that a compact Riemannian manifold with weakly pointwise 1/4-pinched sectional curvatures is either locally symmetric or diffeomorphic to a space form. More generally, we classify all compact, locally irreducible Riemannian manifolds M with the property that M × R ² has non-negative isotropic curvature. The first author was partially supported by a Sloan Foundation Fellowship and by NSF grant DMS-0605223. The second author was partially supported by NSF grant DMS-0604960.     

The homology of irregular dihedral branched covers ofS 3

Partially supported by NSF grant DMS-9208052 and the MSRI NSF grant DMS-9022140. The author held an MSRI Research Professorship while the paper was being written     

Area minimizers in a K3 surface and holomorphicity

No Abstract. . Received: October 2004 Revision: May 2005 Accepted: July 2005 The second author was partially supported by NSF grant DMS-0304587.     

The homology of cyclic branched covers ofS 3

Partially supported by NSF grant DMS-9208052 and the MSRI NSF grant DMS-9022140. The author held an MSRI Research Professorship while the paper was being written     

Aperiodic tilings of the hyperbolic plane by convex polygons

Several aperiodic hyperbolic tiling systems consisting of a single convex tile are constructed. Research partially supported by GIF grant no. G-454.213.06.95 and SFB 343 Bielefeld. The work of the first author was supported in part by NSF grant DMS-9424613. The work of the second author was supported in part by a grant of the Israel Academy of Sciences, and by the Edmund Landau Center for Research in Mathematical Analysis supported by the Minerva Foundation (Federal Republic of Germany).     

On the convergence of subdifferentials of convex functions

Research partially supported by NSF grant DMS-9001096. This author would like to thank l'équipe d'analyse convexe of U.S.T.L. Montpellier for its hospitality.     

Quasiequational Theories of Flat Algebras

We prove that finite flat digraph algebras and, more generally, finite compatible flat algebras satisfying a certain condition are finitely q-based (possess a finite basis for their quasiequations). We also exhibit an example of a twelve-element compatible flat algebra that is not finitely q-based. The first author was partially supported by the grant # 201/02/0594 of the Grant Agency of the Czech Republic, and by the Institutional grant MSM0021620839; the second author was partially supported by the grant No. Tn37877 of the Hungarian National Foundation for Scientific Research (OTHA); the third author was supported by the NSF grant # DMS-9971352.     

Central Value Of Automorphic <Emphasis Type="Italic">L</Emphasis>-Functions

We prove a generalization to the totally real field case of the Waldspurger’s formula relating the Fourier coefficient of a half integral weight form and the central value of the L-function of an integral weight form. Our proof is based on a new interpretation of Waldspurger’s formula as a combination of two ingredients – an equality between global distributions, and a dichotomy result for theta correspondence. As applications we generalize the Kohnen–Zagier formula for holomorphic forms and prove the equivalence of the Ramanujan conjecture for half integral weight forms and a case of the Lindel?f hypothesis for integral weight forms. We also study the Kohnen space in the adelic setting. The first author was partially supported by NSF grant DMS-0070762. The second author was partially supported by NSF grant DMS-0355285. Received: July 2005 Accepted: August 2005     

Model completeness for trivial, uncountably categorical theories of Morley rank 1

We show that if T is a trivial uncountably categorical theory of Morley Rank 1 then T is model complete after naming constants for a model.The second author was partially supported by NSF grant DMS-0300080.     

Gradient estimates,Harnack inequalities and estimates for heat kernels of the sum of squares of vector fields

Research partially supported by NSF grant no. DMS-87-04209 and DMS-90-04062     

On groups of hyperbolic length

We investigate the recently introduced notion of rotation numbers for periodic orbits of interval maps. We identify twist orbits, that is those orbits that are the simplest ones with given rotation number. We estimate from below the topological entropy of a map having an orbit with given rotation number. Our estimates are sharp: there are unimodal maps where the equality holds. We also discuss what happens for maps with larger modality. In the Appendix we present a new approach to the problem of monotonicity of entropy in one-parameter families of unimodal maps. This work was partially done during the first author’s visit to IUPUI (funded by a Faculty Research Grant from UAB Graduate School) and his visit to MSRI (the research at MSRI funded in part by NSF grant DMS-9022140) whose support the first author acknowledges with gratitude. The second author was partially supported by NSF grant DMS-9305899, and his gratitude is as great as that of the first author.     

Bessel identities in the waldspurger correspondence over the real numbers

We prove certain identities between Bessel functions attached to irreducible unitary representations ofPGL ₂(R) and Bessel functions attached to irreducible unitary representations of the double cover ofSL ₂(R). These identities give a correspondence between such representations which turns out to be the Waldspurger correspondence. In the process we prove several regularity theorems for Bessel distributions which appear in the relative trace formula. In the heart of the proof lies a classical result of Weber and Hardy on a Fourier transform of classical Bessel functions. This paper constitutes the local (real) spectral theory of the relative trace formula for the Waldspurger correspondence for which the global part was developed by Jacquet. Research of first author was partially supported by NSF grant DMS-0070762. Research of second author was partially supported by NSF grant DMS-9729992 and DMS 9971003.     

A bilinear oscillatory integral along parabolas

We establish an norm estimate for a bilinear oscillatory integral operator along parabolas incorporating oscillatory factors . The first author was partially supported by NSF grant of China grant 10671079. The second author was supported by NSF grant DMS-0456976.     

Fourier Series Approximation of Linear Fractional Stable Motion

An approximation of the linear fractional stable motion by a Fourier sum is presented. In the continuous sample path case precise error bounds are derived. This approximation method is used to develop a simulation method of the sample path of linear fractional stable motions. The second author was partially supported by NSF grant DMS-0417869.     

Initial segments of the degrees of constructibility

We show that any constructible, constructibly countable, (dual) algebraic lattice is isomorphic to the degrees of constructibility of reals in some generic extension ofL. Research partially supported by NSF grant DMS-8601777. Research partially supported by NSF grant DMS-8601048 and grant 84-00067 from the U.S.-Israel Binational Science Foundation. Thanks are also due to Uri Abraham and Mati Rubin for helpful discussions about initial segments of the degrees of constructibility and to Bill Lampe for information on algebraic lattices.     

Separable pure subgroups of completely decomposable torsion-free abelian groups

Research partially supported by NSF grant no. DMS-8701074.     

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.