Silver‐Triggered Activity of a Heterogeneous Palladium Catalyst in Oxidative Carbonylation Reactions

A silver‐triggered heterogeneous Pd‐catalyzed oxidative carbonylation has been developed. This heterogeneous process exhibits high efficiency and good recyclability, and was utilized for the one‐pot construction of polycyclic compounds with multiple chiral centers. AgOTf was used to remove chloride ions in the heterogeneous catalyst Pd‐AmP‐CNC, thereby generating highly active Pd^II, which results in high efficiency of the heterogeneous catalytic system.     

Silver-Triggered Activity of a Heterogeneous Palladium Catalyst in Oxidative Carbonylation Reactions

A silver-triggered heterogeneous Pd-catalyzed oxidative carbonylation has been developed. This heterogeneous process exhibits high efficiency and good recyclability, and was utilized for the one-pot construction of polycyclic compounds with multiple chiral centers. AgOTf was used to remove chloride ions in the heterogeneous catalyst Pd-AmP-CNC, thereby generating highly active Pd^II, which results in high efficiency of the heterogeneous catalytic system.     

Resolvent of the Laplacian on geometrically finite hyperbolic manifolds

For geometrically finite hyperbolic manifolds Γ\ℍ ⁿ⁺¹, we prove the meromorphic extension of the resolvent of Laplacian, Poincaré series, Einsenstein series and scattering operator to the whole complex plane. We also deduce the asymptotics of lattice points of Γ in large balls of ℍ ⁿ⁺¹ in terms of the Hausdorff dimension of the limit set of Γ.     

The Yamabe problem on stratified spaces

We introduce new invariants of a Riemannian singular space, the local Yamabe and Sobolev constants, and then go on to prove a general version of the Yamabe theorem under that the global Yamabe invariant of the space is strictly less than one or the other of these local invariants. This rests on a small number of structural assumptions about the space and of the behavior of the scalar curvature function on its smooth locus. The second half of this paper shows how this result applies in the category of smoothly stratified pseudomanifolds, and we also prove sharp regularity for the solutions on these spaces. This sharpens and generalizes the results of Akutagawa and Botvinnik (GAFA 13:259–333, 2003) on the Yamabe problem on spaces with isolated conic singularities.     

Singular Solutions of Fractional Order Conformal Laplacians

We investigate the singular sets of solutions of a natural family of conformally covariant pseudodifferential elliptic operators of fractional order, with the goal of developing generalizations of some well-known properties of solutions of the singular Yamabe problem.     

Nonresonant third harmonic generation in thallium vapour

Feasibility of nonresonant third harmonic generation in thallium is investigated. The third order susceptibility, for driving frequencies in the visible region, is calculated and the phase matching with Ar as buffer gas is investigated. Other related quantities like coherence length, minimum pulse lengths required to get phase matching and power input requirements to achieve 50% conversion are also calculated.     

Curvature and uniformization

We approach the problem of uniformization of general Riemann surfaces through consideration of the curvature equation, and in particular the problem of constructing Poincaré metrics (i.e., complete metrics of constant negative curvature) by solving the equation Δu-e ^2u=K_o(z) on general open surfaces. A few other topics are discussed, including boundary behavior of the conformal factore ^2u giving the Poincaré metric when the Riemann surface has smoothly bounded compact closure, and also a curvature equation proof of Koebe's disk theorem. Research supported in part by NSF Grant DMS-9971975 and also at MSRI by NSF grant DMS-9701755. Research supported in part by NSF Grant DMS-9877077     

Moduli Spaces of Singular Yamabe Metrics

Complete, conformally flat metrics of constant positive scalar curvature on the complement of points in the -sphere, , , were constructed by R. Schoen in 1988. We consider the problem of determining the moduli space of all such metrics. All such metrics are asymptotically periodic, and we develop the linear analysis necessary to understand the nonlinear problem. This includes a Fredholm theory and asymptotic regularity theory for the Laplacian on asymptotically periodic manifolds, which is of independent interest. The main result is that the moduli space is a locally real analytic variety of dimension . For a generic set of nearby conformal classes the moduli space is shown to be a -dimensional real analytic manifold. The structure as a real analytic variety is obtained by writing the space as an intersection of a Fredholm pair of infinite dimensional real analytic manifolds.