RUI: Noncommutative Geometry: Curvature and Rigidity of Noncompact Manifolds

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AbstractAward: DMS-0405867Principal Investigator: Stanley S. ChangThe main projects funded by this award explore the existenceproblem for Riemannian metrics of positive scalar curvature onnoncompact manifolds. The work of Gromov and Lawson on thecompact case of this problem has stimulated important parts ofthe development of noncommutative geometry and controlledtopology and these projects are formulated within the context ofthe stable Gromov-Lawson-Rosenberg conjecture. The principalinvestigator and collaborators aim to construct an appropriateassembly map in the noncompact case, to test that theory innatural examples, and to study with the generalized Roe algebrathe coarse quasi-isometry type of manifolds that do admitcomplete metrics of positive scalar curvature. Noncompactmanifolds obtained by deleting submanifolds from larger compactmanifolds and non-Galois or irregular covering spaces of compactmanifolds are promising sources of examples for study, andnoncompact manifolds of finite asymptotic dimension will receiveparticular attention. Other directions of investigation includethe zero-in-the-spectrum conjecture and expander graphs.Noncommutative geometry is an approach to the study of geometricobjects through algebras of natural functions and operators. Forexample, much of the geometry of a sphere is captured insolutions to equations that model vibration, and by thedifferential operators that appear in those equations. Roundspheres are examples of manifolds of positive curvature, the flatplane has curvature zero, and the saddle point in a mountain passis a model for space of negative curvature. Mathematicians havefound a number of tests that must be passed by any space thatwould be a candidate to carry a geometry of positive curvature,and these projects will advance that effort. Another line ofwork described in the proposal concerns expander graphs, whichwere originally introduced in computer science as models of largenetworks with good communication properties but have also turnedout to be a source of potential counterexamples for geometricquestions. These will be the subject of seminars involvingstudents and faculty from Wellesley College and nearbyinstitutions.