# Noncommutative Geometry

Noncommutative Geometry extends the natural correspondence between geometric spaces and commutative algebras, to the **noncommutative** case. This extension involves an algebraic reformulation of the classical tools, such as measure theory, topology, differential calculus and Riemannian geometry, to the noncommutative situation.

The existence of many natural spaces for which the classical set-theoretic tools of analysis, such as measure theory, topology, calculus, and metric ideas lose their pertinence, but which correspond very naturally to a noncommutative algebra. Such spaces arise both in mathematics and in **quantum physics**: the space of Penrose tilings, of irreducible unitary representations of a discrete group, and the phase space in quantum mechanics.

## Research Interests of the Group

One of the main objects of study are the Operator algebras, that includes C*-algebras and von Neumann algebras. While C*-algebras can be seen as a generalization of topological spaces, von Neumann algebras are from Measure spaces. Our research on Operator algebras covers topics as:

- Classification of C*-algebras,
- dynamical systems,
- representation theory of locally compact groups and quantum groups,
- ergodic theory,
- wavelets.

Other research interests are to explore connections between noncommutative geometry, applied harmonic analysis and quantum mechanics, where different aspects of the **Heisenberg group** is in some sense the unifying theme. Also the link between noncommutative geometry and signal analysis, where we study the projective modules over **noncommutative tori** from the perspective of time-frequency analysis.