Concerning the existence of Einstein and Ricci soliton metrics on solvable Lie groups

In this work we investigate solvable and nilpotent Lie groups with special metrics. The metrics of interest are left-invariant Einstein and algebraic Ricci soliton metrics. Our main result shows that the existence of a such a metric is intrinsic to the underlying Lie algebra. More precisely, we show how one may determine the existence of such a metric by analyzing algebraic properties of the Lie algebra in question and infinitesimal deformations of any initial metric. Our second main result concerns the isometry groups of such distinguished metrics. Among the completely solvable unimodular Lie groups (this includes nilpotent groups), if the Lie group admits such a metric, we show that the isometry group of this special metric is maximal among all isometry groups of left-invariant metrics. We finish with a similar result for locally left-invariant metrics on compact nilmanifolds.Comment: 28 page

A step towards the Alekseevskii Conjecture

We provide a reduction in the classification problem for non-compact, homogeneous, Einstein manifolds. Using this work, we verify the (Generalized) Alekseevskii Conjecture for a large class of homogeneous spaces.Comment: 12 pages, proof of main result simplified, final version to appear in Math. Annale

Improving Teen Engagement in Buffalo and Erie County

Erie County offers many opportunities for teen engagement through afterschool programs, summer jobs programs, and internships. National research reveals many benefits from these programs, such as improved academic achievement, better health outcomes, and reductions in violence. Participation, however, often drops significantly after youth enter high school. A better understanding of adolescent development and youth engagement strategies, ranging from active participation to shared leadership, will draw more teens into programs and maximize their impact