Spectral Evolution Models for the Next Decade

Spectral evolution models are a widely used tool for determining the stellar content of galaxies. I provide a review of the latest developments in stellar atmosphere and evolution models, with an emphasis on massive stars. In contrast to the situation for low- and intermediate mass stars, the current main challenge for spectral synthesis models are the uncertainties and rapid revision of current stellar evolution models. Spectral libraries, in particular those drawn from theoretical model atmospheres for hot stars, are relatively mature and can complement empirical templates for larger parameter space coverage. I introduce a new ultraviolet spectral library based on theoretical radiation-hydrodynamic atmospheres for hot massive stars. Application of this library to star-forming galaxies at high redshift, i.e., Lyman-break galaxies, will provide new insights into the abundances, initial mass function and ages of stars in the very early universe.Comment: 8 pages, to appear in IAU Symp. 262, Stellar Populations - Planning for the Next Decade, eds. G. Bruzual & S. Charlo

Graphs of bounded degree and the pp-harmonic boundary

Let $p$ be a real number greater than one and let $G$ be a connected graph of bounded degree. In this paper we introduce the $p$-harmonic boundary of $G$. We use this boundary to characterize the graphs $G$ for which the constant functions are the only $p$-harmonic functions on $G$. It is shown that any continuous function on the $p$-harmonic boundary of $G$ can be extended to a function that is $p$-harmonic on $G$. Some properties of this boundary that are preserved under rough-isometries are also given. Now let $\Gamma$ be a finitely generated group. As an application of our results we characterize the vanishing of the first reduced $\ell^p$-cohomology of $\Gamma$ in terms of the cardinality of its $p$-harmonic boundary. We also study the relationship between translation invariant linear functionals on a certain difference space of functions on $\Gamma$, the $p$-harmonic boundary of $\Gamma$ with the first reduced $\ell^p$-cohomology of $\Gamma$.Comment: Give a new proof for theorem 4.7. Change the style of the text in the first two section

The first LpL^p-cohomology of some finitely generated groups and pp-harmonic functions

Let $G$ be a finitely generated infinite group and let $p > 1$. In this paper we make a connection between the first $L^p$-cohomology space of $G$ and $p$-harmonic functions on $G$. We also describe the elements in the first $L^p$-cohomology space of groups with polynomial growth, and we give an inclusion result for nonamenable groups

The pp-Royden and pp-harmonic boundaries for metric measure spaces

Let $p$ be a real number greater than one and let $X$ be a locally compact, noncompact metric measure space that satisfies certain conditions. The $p$-Royden and $p$-harmonic boundaries of $X$ are constructed by using the $p$-Royden algebra of functions on $X$ and a Dirichlet type problem is solved for the $p$-Royden boundary. We also characterize the metric measure spaces whose $p$-harmonic boundary is empty.Comment: Added an extra condition to Theorem 1.