Colloquium

I will explain briefly why physicists want to make such statements, and then I will discuss a way that I have developed for trying to make precise these statements. This involves defining the notion of compact quantum metric spaces, and then showing in what way matrix algebras can be viewed as examples of compact quantum metric spaces. This also then involves defining the notion of quantum Gromov-Hausdorff distance between compact quantum metric spaces. The techniques used in applying this to Matrix algebras converge to the sphere involve Berezin quantization and coherent states. I will also try to say a few words about superstructure such as vector bundles.