Indexed squares

We study some combinatorial principles intermediate between square and weak square. We construct models which distinguish various square principles, and show that a strengthened form of weak square holds in the Prikry model. Jensen proved that a large cardinal property slightly stronger than 1-extendibility is incompatible with square; we prove this is close to optimal by showing that 1-extendibility is compatible with square. First author partially supported by NSF grants DMS-9703945 and DMS-0070549. Second author partially supported by NSF Grants DMS-9305990, DMS-9712580, DMS-9996280 and DMS-0088948.     

The subgaussian constant and concentration inequalities

We study concentration inequalities for Lipschitz functions on graphs by estimating the optimal constant in exponential moments of subgaussian type. This is illustrated on various graphs and related to various graph constants. We also settle, in the affirmative, a question of Talagrand on a deviation inequality for the discrete cube. Research supported in part by NSF Grant No. DMS-0405587 and by EPSRC Visiting Fellowship. Research supported in part by NSF Grant No. DMS-9803239, DMS-0100289. Research supported in part by NSF Grant No. DMS-0401239.     

Perturbation of Transition Functions and a Feynman-Kac Formula for the Incorporation of Mortality

Markov transition kernels are perturbed by output kernels with a special emphasis on building mortality into structured population models. A Feynman-Kac formula is derived which illustrates the interplay of mortality with a Markov process associated with the unperturbed kernel. partially supported by NSF grants DMS-0314529 and SES-0345945 partially supported by NSF grants DMS-9706787 and DMS-0314529     

Morera and mean-value type theorems in the hyperbolic disk

Some Morera and mean-value type theorems are proved in the hyperbolic disk. Partially supported by NSF grants DMS-9000619 and CDR-8803012. Partially supported by NSF grants DMS-8703072 and DMS-9000619, and DGI-CYT grant PB 89-0311.     

Separating Cycles in Doubly Toroidal Embeddings

 We show that every 4-representative graph embedding in the double torus contains a noncontractible cycle that separates the surface into two pieces. As a special case, every triangulation of the double torus in which every noncontractible cycle has length at least 4 has a noncontractible cycle that separates the surface into two pieces. Received: May 22, 2001 Final version received: August 22, 2002 RID="*" ID="*" Supported by NSF Grants Numbers DMS-9622780 and DMS-0070613 RID="†" ID="†" Supported by NSF Grants Numbers DMS-9622780 and DMS-0070430     

A hyperbolic surface with a square grid net

We prove the existence of a hyperbolic surface spread over the sphere for which the projection map has all its singular values on the extended real line and such that the preimage of the extended real line under the projection map is homeomorphic to the square grid in the plane. This answers a question raised by è. B. Vinberg. Supported by the Alexander von Humboldt Foundation. Supported by NSF grants DMS-0400636, DMS-0244421, and DMS-0244547.     

The ergodic theorem with time compression

Research supported in part by NSF Grants DMS-8400730 and DMS-8620132.     

Properties of compact complex manifolds carrying closed positive currents

We show that a compact complex manifold is Moishezon if and only if it carries a strictly positive, integral (1, 1)-current. We then study holomorphic line bundles carrying singular hermitian metrics with semi-positive curvature currents, and we give some cases in which these line bundles are big. We use these cases to provide sufficient conditions for a compact complex manifold to be Moishezon in terms of the existence of certain semi-positive, integral (1,1)-currents. We also show that the intersection number of two closed semi-positive currents of complementary degrees on a compact complex manifold is positive when the intersection of their singular supports is contained in a Stein domain. The first author was partially supported by National Science Foundation Grant Nos. DMS-8922760 and DMS-9204273. The second author was partially supported by National Science Foundation Grant Nos. DMS-9001365 and DMS-9204037.     

On the classical Dirichlet problem in the plane with rational data

We consider the Dirichlet problem for the Laplace operator with rational data on the boundary of a planar domain. Our main results include a characterization of the disk as the only domain for which all solutions are rational and a characterization of the simply connected quadrature domains as the only ones for which all solutions are algebraic of a certain type. The first three authors were partially supported by the NSF grants DMS-0305958, DMS-0401215 and DMS-0139008 respectively.     

Countably determined sets and a conjecture of C.W. Henson

Partially supported by NSF grants DMS-9208071 and DMS-9100383     

On the complexity of approximating the maximal inscribed ellipsoid for a polytope

We give a new polynomial bound on the complexity of approximating the maximal inscribed ellipsoid for a polytope.Research supported by NSF Grant DMS-8706133.Research supported by NSF Grant DMS-8904406.     

On the scaling limit of a singular integral operator

The scaling limit and Schauder bounds are derived for a singular integral operator arising from a difference equation approach to monodromy problems. Research supported in part by National Science Foundation grants DMS-02-45371 and DMS-04-05519.     

Minimizers near the first critical field for the nonself-dual Chern–Simons–Higgs energy

The Chern–Simons–Higgs energy serves as a model for high temperature superconductivity. We show the existence of weak solutions to the CSH equations that are minimizers of the CSH energy. The solutions are vortexless for an applied magnetic field h _ex below the critical field strength, whereas vortices appear when h _ex exceeds the critical field strength. D. Spirn was supported in part by NSF grants DMS-0510121 and DMS-0707714. X. Yan was supported in part by NSF grants DMS-0700966 and DMS-0401048.     

Compactness of isospectral conformal metrics onS 3

Research supported by NSF grants DMS-88-16321 and DMS-87-02871.     

Nonisomorphic algebraic models of a smooth manifold

The second author was supported by NSF grants DMS-8905538 and DMS-9096119     

A Morera type theorem forL 2 functions in the Heisenberg group

We obtain a Morera type characterization for the square integrable CR functions on the Heisenberg group. Partially supported by NSF Grant DMS-90524569. $Partially supported by NSF Grant DMS-9000968.     

Finite rank operators with large trace

We give some estimates for how well the normalized trace functional exposes the identity map on a finite-dimensional space. Research supported by NSF Grant DMS-8320632. Research supported by NSF Grant DMS-8400960.     

On the vanishing of extensions of modules over reduced enveloping algebras

Supported by NSF grants DMS-9001689 and DMS-9301929     

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.     

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