Focused Research Group (FRG): Fourier analytic and probabilistic methods in geometric functional analysis and convexity
Our
Mission:
The aim of this project is to bring
together tools from Fourier analysis, affine convex geometry, geometric
functional analysis, probability theory, and combinatorics to attack
problems arising in geometry, analysis, and in various areas of applied
mathematics and computer science. On the technical level, the focus is
on the study of properties of (generally high-dimensional) convex
bodies, random matrices, Gaussian measures and processes, and of
approximation problems. Specific directions of planned research are
related to sections and projections of convex bodies, combinatorial dimensions of classes of
functions, singular numbers of random matrices, signal
reconstruction (notably, compressed sensing), and links to quantum
information theory. A combined, focused effort is expected to bring new
insights toward a better understanding of the participants' respective
fields of research, which - while related and occasionally overlapping
- are not identical and often employ different perspectives.
The area of mathematics encompassing
the methods and the problems described above has recently entered a
period of rapid growth. In large part this is due to numerous links to
other fields such as computer science and mathematical physics. In a
nutshell, the wealth of connections between high-dimensional convexity
and applications is due to the complexity of the systems (e.g.,
physical, biological or economical) that one wants to analyze: the
large number of free parameters in such systems may be reflected in the
large dimension of the mathematical object that serves as a model.
Additionally, many results in, say, geometric functional analysis, can
be presented as statements about the complexity of high dimensional
objects in presence of convexity; this explains the links to computer
science.
A major component of this project is
the training of postdocs and graduate students in an integrated
research environment. The dynamic growth of the area and wealth of
applications makes it an ideal topic of study for graduate students and
young researchers. Special attention will be paid to recruiting members
of groups under-represented in the field of mathematics.
This project is supported by NSF DMS grant 0652617,0652571,0652672,0652684,0652722.