Independent Research Fund Denmark - Classical and Quantum Distances

Independent Research Fund Denmark, Research Project 1.


The notion of a space is among the most fundamental concepts in modern mathematics and forms the foundation of a diverse number of central mathematical disciplines, such as geometry, topology, analysis and algebra. The common idea within all these disciplines, is that a space is a collection of points endowed with some additional structure: if we require there to be a notion of distance between the points we end up with a metric space, if we require there to be a way of multiplying points we obtain a
group, and if we demand that there be a notion of tangents, we arrive at a geometric object known as a manifold. It is often the case that very interesting theories, with important applications in physics and chemistry, appear when the above mentioned notions are suitably mixed: for instance, by combining a group structure with a manifold structure a so-called Lie group emerges, and these are exactly the objects governing the symmetries in classical physics. Although classical spaces are very well suited
for describing classical physics, the discovery of quantum mechanics in the early 20th century made it clear that such spaces could not fully describe the quantum mechanical phenomena; what is needed instead are certain non-commutative algebras, which substitute the algebra of classical observables
consisting of functions on a (classical) space. One often thinks of such a non-commutative algebra as being an “algebra of functions” on a (non-existing!) quantum space, and this point of view has turned out tremendously successful and has given rise to a lot of deep mathematical insights. The aim of the project described below is to answer a number of important open questions regarding the metric properties of both classical and quantum spaces, which will shed new light on classical open
problems and unveil exciting new aspects of the rapidly evolving theory of quantum metric spaces.