Bachelor and master projects, Franz Luef

Here is a list of potential topics for bachelor projects and master projects. Please contact me if you have any further inquiries contact me. In case you are interested in analysis and its connections to quantum physics, signal analysis, or machine learning, but the listed projects are not quite what you are looking for, just get in touch with me and we can try to meet and figure out a project that suits your background and wishes.

In this theory functions are replaced by a special class of operators, so-called trace class operators, since for these one can define an analog of an integral for functions, much in the same way as the trace of a matrix in linear algebra. There is also a variant of a transform for operators that behaves like the Fourier transform of operators, the so-called Fourier-Wigner transform. In addition, there are convolutions between functions and operators, as between two operators, which are turned under the Fourier-Wigner transform into products of Fourier transforms, very much like in the case of functions.

The origins of this theory go back to the seminal work of Reinhard F. Werner on quantum harmonic analysis in 1984 and were motivated by questions in quantum physics. In a series of papers in collaboration with Eirik Skrettingland we have started to develop a link between Werner's work and signal analysis/time-frequency analysis (see references). We are currently working on consequences for machine learning, in particular convolutional neural networks, but also on extensions to other settings.

Here is a list of topics related to this circle of ideas.

Potential topics:

• From harmonic analysis to quantum harmonic analysis: In order to get an idea of the quantum aspect of this theory it is of interest to understand the counterpart of some inequalities and theorems in harmonic analysis: Riemann-Lebesgue's Lemma, Young's inequality etc.
• Structure of systems of signals/functions/data sets: Based on the accumulated Cohen class associated with a system of signals/functions/data sets one can extract the relevant information contained in this system of signals/functions/data sets. Applying this idea to a concrete data set/functions might be a worthwhile undertaking.
• Back to quantum mechanics: Using the results developed in the references it might be worth exploiting its consequences for Werner's original motivation, quantum mechanics.
• Link between quantum harmonic analysis and complex analysis: Since quantum harmonic analysis contains the theory of localization operators, which as a special case contains Toeplitz operators on Bargmann-Fock spaces, a discussion of the complex analysis aspects might be a nice project.

References

1. Luef, Franz; Skrettingland, Eirik. Mixed-State Localization Operators: Cohen’s Class and Trace Class Operators. Journal of Fourier Analysis and Applications; Volume 25(4), 2064-2108, 2019.
2. Luef, Franz; Skrettingland, Eirik. On Accumulated Cohen's Class Distributions and Mixed-State Localization Operators. Constructive approximation, to appear.
3. Luef, Franz; Skrettingland, Eirik. Convolutions for localization operators. Journal des Mathématiques Pures et Appliquées;Volum 118. s. 288-316, 2018.
4. Werner, Reinhard. Quantum harmonic analysis on phase space. Journal of mathematical physics 25.5: 1404-1411, 1984.

There is a quite intriguing link between signal analysis and some classes of operator algebras, the Moyal plane and the noncommutative torus. These two are examples of analogs of Riemannian manifolds in the setting of operator algebras. It turns out that the study of Morita equivalences for a noncommutative torus leads to the same structures as tools designed by electrical engineers to study and analyze audio signals.

Topics in this area are of a broad nature, ranging from problems based on Fourier analysis to algebra, and are based on the following papers.

Potential topics:

• Finite-dimensional setting: Motivated by applications in signal analysis and physics there is a theory that discusses noncommutative tori in the finite-dimensional setting. In this project, we will develop these aspects in detail and address problems in physics or time-frequency analysis.
• Structured systems of functions: The theory of operator algebras provides an interesting way to generate systems of functions with structure, via the theory of Hilbert C*-modules, which are a natural extension of Hilbert spaces. In this project, we investigate some examples related to this circle of ideas.

References

1. Chakraborty, Sayan; Luef, Franz. Metaplectic transformations and finite group actions on noncommutative tori. Journal of operator theory 2019 ;Volume 821) s. 147-172.
2. Jakobsen, Mads Sielemann; Luef, Franz. Sampling and periodization of generators of Heisenberg modules. International Journal of Mathematics 2019 ;Volume 30(10) s. 1950051
3. Enstad, Ulrik Bo Rufus; Austad, Are. Heisenberg modules as function spaces. Journal of Fourier Analysis and Applications 2020;Volume 26.