This award supports a Research Network in the Mathematical Sciences. The GEometric structures And Representation varieties (GEAR) network encompasses mathematicians with backgrounds in diverse areas such as classical TeichmÃ¼ller theory, discrete subgroups of Lie groups, 3- manifold topology, harmonic maps, dynamics, and moduli spaces of Higgs bundles and gauge theory. In recent years researchers in these areas have converged on common problems clustered around locally homogeneous structures on manifolds and moduli spaces of representations of finitely generated groups. New techniques and tools in one area have led to advances in another, and the contrasting points of view have highlighted new questions.
The goals of the GEAR network are to spur cross-pollination of ideas from one area to another, to build a research community transcending current divisions, to train students and researchers to cross traditional boundaries, and by these means to advance the mathematics targeted by the network. The main programs of the network include visits and exchanges among researchers at different locations, focused meetings and workshops designed to bridge gaps between targeted subspecialties or to address specific problems, network-wide retreats to help forge a common research community, graduate student internships and summer research experiences, and cyber-networking activities. Spread over 46 locations in the US, Canada and Europe, the nodes of the network cover a wide range of institutions, including several with historically low levels of resources for research. The nodes are arranged around central hubs at The University of Illinois at Urbana-Champaign, The University of Maryland, and Stanford University, with the lead hub in Urbana-Champaign.
GEAR Research Network home page: www.gear.math.illinois.edu