Topics in Quantum Geometry

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AbstractRieffelI propose to develop and apply a theory of quantum metric spaces. My point of departure will be the ideas of Alain Connes concerning quantum Riemannian manifolds. My emphasis will be on extending to quantum spaces, in the operator algebras setting, the notion of the Gromov-Hausdorff distance between metric spaces. I expect to apply this theory to the many situations already in view in the physics and mathematics of quantization where one has a sequence of quantum spaces which appear to be converging to another space, either quantum or classical. In most such situations of which I am aware, the present notions of convergence are either just heuristic, or quite weak. I believe that a quantum Gromov-Hausdorff convergence can often be applied to give a stronger form of convergence. Examples range from approximation by finite dimensional algebras such as frequently occurs in Berezin-Toeplitz quantization, to approximation of quantum field-theory models, especially those of integrable systems, say by lattice models, all the way to the hypothesized "M-theory" of string theory and its (hypothesized) approximation by "Matrix theory".Our nation's technological and economic success has at its foundation the mathematical models of the world around us which scientists develop in order to understand how to use the flood of data which flows from the laboratories of the experimental scientists. But human beings and computers can only deal with finite collections of numbers at a time. Thus in applying these mathematical models it is almost always necessary to approximate the infinite variability of our world by finite collections of numbers. It is then crucial to understand how valid any given approximation is. With respect to individual calculations this matter has received extensive study. But less study has been made of how complex models as a whole can be approximated well by simpler models as a whole. Relatively little is known about such "global" approximations in the case of the models of quantum physics, which is the part of physics which governs chemical and biochemical reactions, the functioning of semi-conductors, and many other key technologies. I propose to extend to the quantum realm one form of global approximation, called Gromov-Hausdorff distance, which has in recent years proved to be quite powerful in the classical realm for dealing with certain broad classes of mathematical situations. I expect to apply this global approximation method to better understand how to effectively approximate various models of quantum phenomena of current interest.