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.     

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.     

On the Shintani zeta function for the space of binary quadratic forms

Partially supported by NSF Grant DMS-8803085, DMS-8610730     

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

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

The ergodic theorem with time compression

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

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.     

Compactness of isospectral conformal metrics onS 3

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

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     

On the vanishing of extensions of modules over reduced enveloping algebras

Supported by NSF grants DMS-9001689 and DMS-9301929     

Working below a low2 recursively enumerably degree

Research partially supported by NSF Grants DMS-861018 and DMS-8902797     

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     

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     

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     

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.     

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.     

Rank two vector bundles over regular elliptic surfaces

Research partially supported by NSF grants DMS-85-03743 and DMS-87-03569 and the Alfred P. Sloan Foundation     

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.     

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.     

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.