During the past few years the principal investigator have been developing a theory of quantum metric spaces, that is, of quantum spaces equipped with the analog of the notion of a distance (i.e., metric) on an ordinary space. He does this within the setting of algebras of operators on Hilbert space, which is the setting used by physicists to model quantum systems. A central part of his theory consists of a quantum analog of the classical Gromov-Hausdorff distance between metric spaces. He has given several applications of these ideas, notably to the convergence of matrix algebras to coadjoint orbits of compact Lie groups, which is one of the main examples of interest to physicists. Recently he has had initial success in extending his notion of quantum Gromov-Hausdorff distance so as to treat the analogs of vector bundles over quantum metric spaces. (Vectors bundles provide the foundation for the gauge theories widely used by physicists.) He has also had recent success going in the direction of expressing the quantum distances in terms of Dirac operators, with the technical advantages that Dirac operators bring. In this project he will continue to strengthen this theory and to apply it in several directions suggested by the many situations in the physics and mathematics of quantum models where one has a sequence of quantum spaces that appears to converge to another space, either quantum or classical. In particular, he will try to develop analogs of quantum Gromov-Hausdorff convergence for quantum versions of superstructures beyond vector bundles, such as for connections, Dirac operators, and Yang-Mills actions. He hopes, as well, to begin treating dynamical issues.
Our nation's technological and economic success has at its foundation the mathematical models of the world around us. Scientists develop such models in order to understand how to use the flood of data that flows from the laboratories of the experimental scientists. However, human beings and computers can deal only 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. On the other hand, 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. Since quantum physics is the part of physics that governs chemical and biochemical reactions, the functioning of semiconductors, and many other key technologies, it is of great importance to understand how well complex quantum models can be approximated by simpler models. In the classical realm, the global notion of approximation known as "Gromov-Hausdorff distance" is widely used. The principal investigator has developed a quantum analog of Gromov-Hausdorff distance and successfully applied it to a few significant examples. In this project he seeks to strengthen this theory and to apply the strengthened theory to a broader class of examples. This should lead to a better understanding of how to approximate in an effective manner various models of quantum phenomena of current importance.
