Developing Effective K-16 Geoscience Partnerships

This article describes the benefits of research partnerships to scientists, students, and teachers. There is growing awareness that the way science is experienced in the K-16 classroom deviates greatly from the experiences of practicing researchers. Whereas researchers are immersed in more open-ended observation and inquiry, many K-16 students find themselves cramming to memorize core scientific content in preparation for standardized examinations. This issue can be mitigated by the development of partnerships in which scientists benefit by added human resources (teachers and students) for data collection and analysis, and teachers and students benefit from a learning process that fosters creativity, sets high standards, teaches problem solving, and is highly motivating. Educational levels: Graduate or professional

Limit sets for modules over groups on CAT(0) spaces -- from the Euclidean to the hyperbolic

The observation that the 0-dimensional Geometric Invariant $\Sigma ^{0}(G;A)$ of Bieri-Neumann-Strebel-Renz can be interpreted as a horospherical limit set opens a direct trail from Poincar\'e's limit set $\Lambda (\Gamma)$ of a discrete group $\Gamma$ of M\"obius transformations (which contains the horospherical limit set of $\Gamma$) to the roots of tropical geometry (closely related to $\Sigma ^{0}(G;A)$ when G is abelian). We explore this trail by introducing the horospherical limit set, $\Sigma (M;A)$, of a G-module A when G acts by isometries on a proper CAT(0) metric space M. This is a subset of the boundary at infinity of M. On the way we meet instances where $\Sigma (M;A)$ is the set of all conical limit points, the complement of a spherical building, the complement of the radial projection of a tropical variety, or (via the Bieri-Neumann-Strebel invariant) where it is closely related to the Thurston norm.Comment: This is the final published versio