Mathematical Sciences: Index Theory, Coarse Geometry and Topology of Manifolds

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Fox NSF/CBMS Regional Research Conference Proposal #9414820 ABSTRACT: A fundamental goal of mathematics is to find the RinvariantsS of objects studied---the characteristic signatures that distinguish these objects from others. For example, a ball is distinguished from a donut by the fact that a donut has one hole while the ball has none. The NSF CBMS program, RIndex Theory, Coarse Geometry, and Topology of Manifolds, S will give an overview of current work in finding invariants for a class of geometric objects. The type of invariants studied in this research are similar in spirit to those in the donut/ball example, though vastly more complicated and useful. There is already a hint that this research could have profound applications in the exploding field of biotechnology. For example, index theory gives biochemists the tools to describe and classify the various ways that the macro-molecules of life twist and turn in space, a difficult and extremely important problem. John Roe of Oxford University will give ten lectures on this multifaceted area of mathematics. The lectures will serve several purposes. They will present new and difficult ideas that blend several areas of mathematics, including topology, geometry and analysis. They will also introduce graduate students to ground-breaking and fundamental mathematical research. Additionally, John Roe, a gifted writer on mathematics, will produce a monograph on these lectures, which will prove a valuable resource for those attending, and for the mathematics community at large.