ESI12

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




Four topics will be covered:
(A) Phase space methods for pseudodifferential operators
Team: K. Gröchenig, M. de Hoop, M. Ruzhansky
The application of timefrequency methods is one of the hot topics in the field, with
very interesting research efforts 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
differential equations (like the Schröodinger equation) using nonorthogonal expansions,
Gaborframe representations of operators, localization principles, noncommutative version
of Wiener's theorem and others.

(B) Numerical Methods in TimeFrequency 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 timefrequency 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 nonorthogonal 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 TimeFrequency Methods
Team: F. Luef, M.A. Rieffel, 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. Rieel in Berkeley. He has already established some very interesting connections
between the concrete problems coming up in TFanalysis, 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 efforts, while M. de Hoop will take
care of the connections to geophysical exploration and microlocal analysis. M. Ruzhansky
will push the lines of research connecting pseudodifferential operators with group
representation theory.

(D) TimeFrequency 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 TFmethods, applied
to signals. In fact, many methods used for the processing of audio signals (such as
MP3) are making direct use of the shorttime 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 firstnamed scientists, and was the coordinator
of the HASSIP RTN (EU) network (20022006). 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 TFmethods 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 (20082011,
so far) on "Computational Engineering" and the recently started cooperation with ESO
(European Southern Observatories) has to be mentioned. NuHAG is contribution an
inkind 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 TFcommunity.


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