The Novikov and Exactness Conjectures for Discrete Groups

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AbstractGuentnerThe proposed research comprises two distinct projects; therelationship between the Novikov and exactness conjectures fordiscrete groups, and the use of global analytic techniques to studyquantum mechanical systems. The first is motivated by the observationthat the classes of groups for which the Novikov and exactnessconjectures are known coincide to a large degree; both classes containamenable groups, hyperbolic groups and Coxeter groups, for example.This project is joint with J. Kaminker. The second is based on theidea that the Berezin-Toeplitz quantization can be analyzed usingspectral properties of family of Dirac-type operators. This projectis joint with J. Trout.The Novikov conjecture, one of the most important problems intopology, has stimulated a tremendous amount of mathematical researchover the last thirty years. The exactness conjecture is purelyanalytic. That there could exist a connection between these twoconjectures from entirely different branches of mathematics issomewhat surprising, but nevertheless is supported by empiricalevidence. We plan to develop more fully the relationship betweenthese conjectures. This research has bearing on a number of importantoutstanding problems including the Baum-Connes Conjecture. At theheart of the connection between mathematics and physics is the theoryof quantum mechanics. We propose a new method to analyze quantummechanical systems based on geometric properties of certain systems ofdifferential equations. This work should lead to a betterunderstanding of a number of quantum mechanical systems frommathematical physics.