Mathematical Sciences: Differential Topology and Dynamics ofGroup Actions and Foliations

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9401688 Hurder The study of geometric properties of open complete manifolds is a rapidly developing area of research in pure mathematics. Hurder's research program seeks to gain a deeper understanding of geometric and dynamical phenomenon on open manifolds by exploring the "structures at infinity" and asymptotic symmetries of spaces. His investigation combines techniques from coarse cohomology, cyclic cohomology, buildings, dynamics and spectral theory of Dirac-type operators. The structure theory for the asymptotic behavior of actions has become much more sophisticated in recent years, and is an emerging technology of choice for addressing some well-known problems regarding the classification of open manifolds and the symmetry groups on these spaces. The problems addressed in this work are also closely tied to mathematical physics and explaining natural phenomenon, especially the work relating asymptotic structures on open spaces to the spectral theory of geometric operators. Other aspects of the project are currently being applied to the analysis of computer simulations of tiling maps and random dynamics of motions in two and three space, with eventual applications to explaining the behavior of gases (the higher dimensional billiard problem). ***