Spanning trees and a conjecture of Kontsevich

Kontsevich conjectured that the number of zeros over the fieldF _q of a certain polynomialQ _G associated with the spanning trees of a graphG is a polynomial function ofq. We show the connection between this conjecture, the Matrix-Tree Theorem, and orthogonal geometry. We verify the conjecture in certain cases, such as the complete graph, and discuss some modifications and extensions.Partially supported by NSF grant #DMS-9743966.     

Star-linear equational theories of groupoids

We prove that there are precisely six equational theories E of groupoids with the property that every term is E-equivalent to a unique linear term. Presented by J. Berman. Received November 11, 2004; accepted in final form March 12, 2006. The first and third authors were supported by the Ministry of Science and Environment of Serbia, grant no. 144011; the second and fifth authors were supported by MŠMT, research project MSM 0021620839, and by the Grant Agency of the Czech Republic, grant #201/02/0594; the fourth author was supported by the NSF grant #DMS-0245622.     

Positivity and strict contractivity of functions of operators

Carathéodory class functionsf(z) are described having the property that the self-adjoint part off(A) is positive definite for every contractionA whose spectral radius is less than 1. Analogous results are obtained for bounded analytic functions in the unit disc, and for the Nevanlinna class. Applications to Markov chains are indicated.Partially supported by the US Air Force Grant AFOSR-94-0293.Partially supported by the NSF Grant DMS-9500924.     

Flagf-vectors and thecd-index

Partially supported by the Goran Gustafsson Foundation at the Royal Institute of Technology (Kungl. Tekniska H?gskolan), Stockholm, and by NSF grants #DMS-8401376 and #DMS-9206374     

The Capelli identity,the double commutant theorem,and multiplicity-free actions

Partially supported by NSF grant #DMS-8807336     

Polynomial diffeomorphisms ofC 2

Partially supported by NSF grant #DMS-8602020     

Distances between measures from 1-dimensional projections as implied by continuity of the inverse radon transform

Summary Distances between measures on IR ^d are determined from distances between their 1-dimensional projections. The method employed involves considering the 1-dimensional projections to be the Radon transform of the measures. Crucial to the main theorem is a continuity result for the inverse Radon transform. Focus is restricted to the Prohorov, dual bounded Lipschitz and d _k metrics which metrize weak convergence of probability measures. These metrics are related to each other and to the Sobolev norms. The d _k results extend from measures to generalized functions.Partially supported by NSF Grant No. MCS-81-01895Partially supported by NSF Grant No. MCS-82-01627 and support from the Mellon Foundation     

Flat affine spheres in R 3

Nondegenerate affine surfaces in R ³ which are affine spheres and have flat affine metrics are classified. Those spheres which are proper are shown to be equivalent to open subsets of the surface defined by xyz=1 or the surface defined by (x ²+y ²)z=1.Partially supported by NSF Grant DMS 8802664.Partially supported by NSERC Operating Grant A2501.     

Equivariant connected sums of compact self-dual manifolds

Partially supported by NSF grant #DMS-9306950 and UC Riverside grant #5-510000-07427-5     

An Algorithm for Corona Solutions on <Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript> (<Emphasis Type="Italic">D</Emphasis>)

We give an algorithm to find corona solutions in H ^∞ (D) for polynomial input data. Partially supported by NSF Grant DMS-0400307.     

Compactly supported box-spline wavelets

A general procedure for constructing multivariate non-tensor-product wavelets that generate an orthogonal decomposition ofL ²(R)^s,s s≥1, is described and applied to yield explicit formulas for compactly supported spline-wavelets based on the multiresolution analysis ofL ²(R)^s 1≤s≤3, generated by any box spline whose direction set constitutes a unimodular matrix. In particular, when univariate cardinal B-splines are considered, the minimally supported cardinal spline-wavelets of Chui and Wang are recovered. A refined computational scheme for the orthogonalization of spaces with compactly supported wavelets is given. A recursive approximation scheme for “truncated” decomposition sequences is developed and a sharp error bound is included. A condition on the symmetry or anti-symmetry of the wavelets is applied to yield symmetric box-spline wavelets. Partially supported by ARO Grant DAAL 03-90-G-0091 Partially supported by NSF Grant DMS 89-0-01345 Partially supported by NATO Grant CRG 900158.     

Some non-uniformly homeomorphic spaces

in this paper we prove that for 0<p, q≤1 the real F-spacesL _q[0, 1] and ℓ _p are not uniformly homeomorphic. The particular casep=q=1 is due to Enflo and our work is motivated by his. Partially supported by NSF grant #DMS-9104040. The author would like to dedicate this paper to Nancy and Bernie Weston on the occasion of their 40th wedding anniversary.     

Anti-self-dual Hermitian metrics on blown-up Hopf surfaces

Work supported in part by NSF Grant #DMS-8704401     

Slopes of effective divisors on the moduli space of stable curves

Research partially supported by NSF Grant #DMS-84-02209.     

Horocycle flows are loosely Bernoulli

It is proved that horocycle flows associated with transitiveC ²-Anosov flows are loosely Bernoulli with respect to their unique ergodic measures. Partially supported by the Sloan Foundation and NSF Grant MCS74-19388.     

Bernoulli maps of the interval

The ergodic properties of expanding piecewiseC ² maps of the interval are studied. It is shown that such a map is Bernoulli if it is weak-mixing. Conditions are given that imply weak-mixing (and hence Bernoulliness). Partially supported by NSF Grant MCS74-19388 and the Sloan Foundation.     

The Radon-Nikodym theorem for von neumann algebras

Let ϕ be a faithful normal semi-finite weight on a von Neumann algebraM. For each normal semi-finite weight ϕ onM, invariant under the modular automorphism group Σ of ϕ, there is a unique self-adjoint positive operatorh, affiliated with the sub-algebra of fixed-points for Σ, such that ϕ=ϕ(h·). Conversely, each suchh determines a Σ-invariant normal semi-finite weight. An easy application of this non-commutative Radon-Nikodym theorem yields the result thatM is semi-finite if and only if Σ consists of inner automorphisms. Partially supported by NSF Grant # 28976 X. Partially supported by NSF Grant # GP-28737 This revised version was published online in November 2006 with corrections to the Cover Date.     

Cohomology and rigidity of fuchsian groups

We prove the localC ³-rigidity of the standard actions of cocompact lattices in PSL(2,ℝ) on a circle, using the Schwarzian and the duality technique for twisted cocycles. Partially supported by NSF Grant #DMS 9403870.     

Range conditions for the exponential Radon transform

A new approach to the range conditions for the exponential Radon transform is introduced. Partially supported by NSF EPSCOR Grant and Wesley Foundation Grant no. 9012019. Partially supported by NSF Grant 33-1807-231.     

The residual nilpotence of verbal wreath products

We obtain necessary and sufficient conditions for residual nilpotence of verbal wreath products. We study also residual nilpotence of the groups of the formF/V(N), whereF is a free groupN◃F, andV(N) is a verbal subgroup ofN. Dedicated to the memory of Professor Brian Hartley Partially supported by NSF Grant DMS9404178.