CAREER: Finer Coarse Geometry

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The research themes in this proposal are centered in geometric group theory and geometric topology, but have connections to several other fields, including number theory, combinatorics, convex geometry, dynamics, and probability. Geometric group theory is most often focused on quasi-isometry invariants: measurements that are insensitive to bounded (additive and multiplicative) distortion of distances. This allows us to pass between the study of groups and the spaces they act on geometrically, and between different Cayley graphs for finitely generated groups. A unifying theme for the research projects described here is the pursuit of "finer" approaches to asymptotic geometry through the use of more sensitive measurements. In particular, the PI proposes to study large-scale geometric statistics that are not invariant under quasi-isometry. This point of view facilitates the study of randomness and asymptotic density in groups, and of properties of "typical" geodesics in spaces.The ideas in play here have already led the PI and her collaborators to new density results in free abelian groups and the Heisenberg group; new invariants used to further the classification program for right-angled Artin groups; and progress in the study of Teichmueller geometry. Statistical geometry is broadly applicable in a range of settings outside pure mathematics, from computer science to medicine---wherever geometric models are made and long-term probabilistic prediction is required. In addition to the research program, this proposal describes a collection of educational projects: principally, the development of an organizational infrastructure for Research Labs engaging faculty and students from a wide variety of institutions. Each lab will work on a cluster of problems around a coherent mathematical nucleus, with opportunities for mutli-level teaching, learning, exploration, and collaboration. There will be conferences and journal issues coordinated with these Labs, guided by the principles of combining expository soundness with research value and emphasizing collaboration.