Over the past several years I developed a theory of quantum metric spaces, within the setting of algebras of operators on Hilbert space. My theory includes an analog of the classical Gromov-Hausdorff distance between metric spaces. I gave several applications of these ideas, notably to the convergence of matrix algebras to coadjoint orbits of compact Lie groups. I propose to continue to strengthen this theory, and to apply it in several directions suggested by the many situations 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. As a major new direction I will try to develop an analogous theory for the quantum versions of the superstructure of vector bundles, connections, Yang-Mills actions, etc. I will also try to extend my theory beyond the quantum analog of locally compact spaces, so as to attempt to deal with the approximations of quantum field-theory models, especially those of integrable systems, say by quantum lattice models. 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. In the classical realm there is an important form of global approximation called Gromov-Hausdorff distance. I have developed a quantum analog of it, and successfully applied it to a few examples. I propose to strengthen this theory, and to apply it to a broader class of examples, so as to better understand how to effectively approximate various models of quantum phenomena of current interest.
