We show that proving the conjectured sharp constant in a theorem of Dennis Sullivan concerning convex sets in hyperbolic 3-space would imply the Brennan conjecture. We also prove that any conformal map f: D→Ω can be factored as a K-quasiconformal self-map of the disk (withK independent of Ω) and a map g: D→Ω with derivative bounded away from zero. In particular, there is always a Lipschitz homeomorphism from any simply connected Ω (with its internal path metric) to the unit disk.

LetX(-ϱB^m×Cⁿ be a compact set over the unit sphere ϱB^m such that for each z∈ϱB^m the fiber X_z={ω∈Cⁿ;(z, ω)∈X} is the closure of a completely circled pseudoconvex domain in Cⁿ. The polynomial hull $\hat X$ of X is described in terms of the Perron-Bremermann function for the homogeneous defining function of X. Moreover, for each point (z₀, w₀)∈Int $\hat X$ there exists a smooth up to the boundary analytic disc F:Δ→B^m×Cⁿ with the boundary in X such that F(0)=(z₀, w₀).

A local-global principle is shown to hold for all conjugacy classes of any inner form of GL(n), SL(n), U(n), SU(n), and for all semisimple conjugacy classes in any inner form of Sp(n), over fields k with vcd(k)≤1. Over number fields such a principle is known to hold for any inner form of GL(n) and U(n), and for the split forms of Sp(n), O(n), as well as for SL(p) but not for SL(n), n non-prime. The principle holds for all conjugacy classes in any inner form of GL(n), but not even for semisimple conjugacy classes in Sp(n), over fields k with vcd(k)≤2. The principle for conjugacy classes is related to that for centralizers.

Cwikel's bound is extended to an operator-valued setting. One application of this result is a semi-classical bound for the number of negative bound states for Schrödinger operators with operator-valued potentials. We recover Cwikel's bound for the Lieb-Thirring constant L_0,3 which is far worse than the best available by Lieb (for scalar potentials). However, it leads to a uniform bound (in the dimension d≥3) for the quotient L_0,d/L^cl_0,d is the so-called classical constant. This gives some improvement in large dimensions.

The harmonic Bergman kernel Q_Ω for a simply, connected planar domain Ω can be expanded in terms of powers of the Friedrichs operator F_Ω ║F_Ω║<1 in operator norm. Suppose that Ω is the image of a univalent analytic function ø in the unit disk with ø' (z)=1+ψ (z) where ψ(0)=0. We show that if the function ψ belongs to a space D_s (D), s>0, of Dirichlet type, then provided that ║ψ║∞<1 the series for Q_Ω also converges pointwise in $\bar \Omega \times \bar \Omega \backslash \Delta (\partial \Omega )$ , and the rate of convergence can be estimated. The proof uses the eigenfunctions of the Friedrichs operator as well as a formula due to Lenard on projections in Hilbert spaces. As an application, we show that for every s>0 there exists a constant C_s>0 such that if ║ψ║_Ds(D)≤C_s, then the biharmonic Green function for Ω=ø (D) is positive.

We establish sharp asymptotics for the L^p-norm of Hermite polynomials and prove convergence in distribution of suitably normalized Wick powers. The results are combined with numerical integration to study an extremal problem on Wiener chaos.

The problem of inversion of the attenuated X-ray transformation is solved by an explicit formula. Several subsequent results are also given.

We give a sharp upper estimate for the response of boundary current-voltage measurements to perturbations of the admittivity in a body that are localized in space and frequency. We calculate the differential of the measurement mapping and study the Gabor symbol of this operator.

We present an elementary and concrete description of the Hilbert scheme of points on the spectrum of fraction rings k[X]_U of the one-variable polynomial ring over a commutative ring k. Our description is based on the computation of the resultant of polynomials in k[X]. The present paper generalizes the results of Laksov-Skjelnes [7], where the Hilbert scheme on spectrum of the local ring of a point was described.

We prove a theorem on non-existence of lacunas of the fundamental solution for hyperbolic differential operators with constant coefficients.