Classical and Quantum Distances

Funded by The Independent Research Fund Denmark
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Short non-scientific description of the project

The notion of a space is one the most central concepts in modern mathematics and forms the foundation of most mathematical disciplines. The basic idea 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 the 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 subtle quantum mechanical phenomena. To remedy this, what is needed 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 is to answer a number of open questions regarding the metric properties of both classical and quantum spaces, which will shed new light on several open problems and unveil exciting new aspects of the rapidly evolving theory of quantum metric spaces.

The project is funded by the Independent Research Fund Denmark and runs from 2019 to 2024.



Research Output






Participants



David Kyed (Professor, PI)

Contact information

Office Phone: +45 - 65504761
E-mail:
Links: Homepage
Publications: arxiv
mathscinet

Research interests

My research is based in the interplay between group theory and operator algebras. Some keywords: L2-Betti numbers, quantum groups, measure equivalence, locally compact groups, amenability, property (T), quantum metric spaces. For more details, see my personal webpage.


Jens Kaad (Associate Professor, co-PI)

Contact information

Office Phone: +45-50789804
E-mail:
Links: Homepage
Publications: arxiv
mathscinet

Research interests

My research focuses on various aspects of noncommutative geometry and in particular on the investigation of finer invariants relating to the differential geometry of noncommutative manifolds. Keywords include: Unbounded KK-theory, Dirac operators, Quantum metric spaces, Algebraic K-theory, Operator spaces.


Konrad Aguilar (Postdoc, 2020)

Contact information

E-mail:
Links: Homepage
Publications: arxiv
mathscinet

Research interests

My research focuses on how the structure of C*-algebras can be displayed in the structure of quantum metric spaces. Keywords: inductive limits, AF algebras, Fell and Jacobson topologies, quantum metric spaces, Gromov-Hausdorff propinquity, compact quantum groups.


Max Holst Mikkelsen (PhD student, 2021-2023)

Contact information

E-mail:

Research interests

Quantum metric spaces
Supervisors: Jens Kaad (main supervisor) and David Kyed (co-supervisor)


Thomas Gotfredsen (PhD student, 2017-2021)

Contact information

E-mail:

Research interests

My current area of interest concerns the interplay between groups and their group cohomologies. Keywords: Coarse Geometry, L2-Betti-numbers, von Neumann Dimension
Supervisor: David Kyed








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