Modern Methods of Time-Frequency Analysis II
September 10th to December 15th, 2012
Erwin Schroedinger Institute (Univ. Vienna)

Four topics will be covered: 

(A) Phase space methods for pseudo-diff erential operators 
Team: K. Gröchenig, M. de Hoop, M. Ruzhansky 
The application of time-frequency methods is one of the hot topics in the field, with very interesting research eff orts going on at the moment. By the time of the planned ESI semester we expect to be able to present exciting new results and summarize the situation, in order to attack even more diffcult problems and find applications to the classical diff erential equations (like the Schröodinger equation) using non-orthogonal expansions, Gabor-frame representations of operators, localization principles, non-commutative version of Wiener's theorem and others.

(B) Numerical Methods in Time-Frequency Analysis 
Team: M. Fornasier, S. Dahlke, H.G. Feichtinger 
The theoretical insight gained in the last years, together with comprehensive MATLAB experiments done at NuHAG, as well as the development of new adaptive methods based on sparse approximation of the solutions of linear equations (as developed by Dahlke, Fornasier and coauthors) give the numerical side of time-frequency analysis more and more importance. Real world problems can be handled, and even the discussion between the connection of insight (approximation, experimental evidence) obtained by computations and the reality of the underlying continuous model deserves more and more fine analysis. The combination of these to aspects (i.e. numerically efficient computations, based on structured, typically non-orthogonal expansions, and the connection between continuous models and finite computations) are the subject of this focus. Again, we see this topic gaining relevance, so that by the time of the ESI semester both relevant fresh results and open problems will be on the table.

(C) Operator Algebras and Time-Frequency Methods 
Team: F. Luef, M.A. Rieff el, H.G. Feichtinger  
The idea of this event exploit the connections between Gabor analysis and noncommutative geometry. The lead in the organization to this section should be given to Franz Luef, who most likely will be ready for habilitation by 2012. He has coauthored a paper with Y. Manin and spends his outgoing Marie Curie Fellowship (2010/2011) with M.A. Rie el in Berkeley. He has already established some very interesting connections between the concrete problems coming up in TF-analysis, and quite abstract constructions (projective modules, Morita equivalences, etc.) in those areas. It appears that this is just the tip of the iceberg. By bringing together leading members from both sides it is very likely that these isolated observations can be turned into a real bridge, allowing also to transfer knowledge between two quite separated (at the moment) disciplines of mathematics. K. Gröchenig will coordinate these eff orts, while M. de Hoop will take care of the connections to geophysical exploration and micro-local analysis. M. Ruzhansky will push the lines of research connecting pseudo-diff erential operators with group representation theory.

(D) Time-Frequency Methods for the Applied Sciences
Team: M. Dörfler, B. Torresani, P. Balazs, H.G. Feichtinger
Musical score is often used as an explanatory model for the use of TF-methods, applied to signals. In fact, many methods used for the processing of audio signals (such as MP3) are making direct use of the short-time Fourier transform. Both M. Dörfler and P. Balazs are presently running interdisciplinary WWTF projects, called the "Audio Miner" (in cooperation with the AI Institute of OEAW) resp. MULAC (realized at ARI, the Acoustic Research Institute of OEAW, in cooperation with NuHAG). B. Torresani is a close cooperation partner of the three first-named scientists, and was the coordinator of the HASSIP RTN (EU) network (2002-2006). The plan for this semester is to bring together people who are coming from the applications and report about problems arising there, and others, who are focusing more on the theoretical foundations (such as the connection to the representation theory of the Heisenberg group, twisted convolution, symplectic Fourier transforms). There are also other areas, where TF-methods are relevant, and which of course partially overlap with some of the other sections. For example, the problem of channel estimation and channel decoding, or problems arising in astronomical image processing. For the first direction the cooperation with the colleagues from TU Vienna (F. Hlawatsch, G. Matz, G. Tauböck) or FTW (T. Zemen), within the SISE NFN network (2008-2011, so far) on "Computational Engineering" and the recently started cooperation with ESO (European Southern Observatories) has to be mentioned. NuHAG is contribution an in-kind project (coordinated by W. Zeilinger, Dept. Astronomy, Univ. Vienna) to Austria's membership, where new mathematical problems will be formulated by 2012. These question are likely to be interesting for the mathematical TF-community.

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