To: jjenkins@nsf.gov Subject: abstract Dear Joe, Here is the abstract you requested. Please let me know if it's what you want. Sorry about the delay in getting this back to you. Because your note came during the holiday break, I put it aside for awhile, and then forgot about it! Thanks for sending the reminder. --Bill TECHNICAL DESCRIPTION This project concerns multivariable operator theory. Specifically, we are concerned with Hilbert modules over the algebra of polynomials in several variables. We seek concrete (numerical) invariants for such objects. The simplest of these is the curvature invariant. The curvature invariant of a Hilbert module H is written K(H). K(H) is analogous to the mean curvature of an even-dimensional Riemannian manifold, and there is an asymptotic formula which allows one to compute its value in many cases. K(H) has turned out to be an integer in all cases we have been able to decide, and in fact it obeys a form of the Gauss Bonnet theorem for the Hilbert modules which admit a "finite free resolution" in the following sense, K(H) = b(1) - b(2) + b(3) - b(4) +.... where b(1), b(2), ... are the Betti numbers of the free resolution. We conjecture that K(H) is always an integer. Not every Hilbert module (in the category we have) has a finite free resolution, and we are developing appropriate tools for computing K(H) in general. GENERAL DESCRIPTION The flow of time in quantum theory is different from the flow of time in classical physics, in that the observable quantities do not commute with each other. For example, Heisenberg's famous equation relating the position and momentum observables of a one-dimensional quantum system is essentially this: PQ - QP = 1. During the past ten years, a small but determined group of mathematicians has been working out the theory of "E_0 semigroups". Among other things, these mathematical objects give the simplest situations which describe the way the flow of time behaves in quantum theory. The current project has grown out of our efforts to find ways of distinguishing between different types of E_0 semigroups. This is accomplished by computing certain numbers associated with them that can take on different values. When one has two E_0 semigoups whose numbers are different, one can be assured that he has two systems which are fundamentally different. This project is concerned with the definition and calculation of numerical quantities which are associated to simpler objects that are closely related to E_0 semigroups.
