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 National Science
Foundation Award #0503642 |
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Topics in Asymptotic Geometric Analysis and its Applications |
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| Investigator(s): |
Stanislaw Szarek (PI)
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| Sponsor: |
Case Western Reserve University, OH 44106 2163684510
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| Start Date/Expiration Date |
2005-06-01 to 2006-05-31 (amended 2005-05-10) |
| Awarded Amount to Date: |
$79,999 |
| Abstract: Abstract
Szarek
This project involves continued research on the geometric,
probabilistic and combinatorial aspects of functional analysis and
convexity theory, the loosely defined area that has been lately
referred to as "asymptotic geometric analysis." Particular attention
will be paid to non-commutative objects and phenomena on one hand
and, on the other hand, to links with other areas of mathematics and
other mathematical sciences, which motivate most of the problems that
are being considered. Sample research topics that are proposed to be
studied include: structural properties of high-dimensional convex
bodies and of high-dimensional normed spaces, metric entropy (of
convex sets or of linear operators) and duality of such entropy and
some geometric questions related to quantum information theory and
quantum. The questions are typically expressed in the language of
local or finite dimensional geometry of Banach spaces and are to be
analyzed using mainly the diverse methods that originated or were
developed in that context.
On an elementary level, Analysis is a study of functions, or
relationships between quantities and the parameters on which they
depend. Since very many naturally appearing relationships are linear
or at least convex, a good understanding of convex functions and,
consequently, of convex sets is a prerequisite for understanding
those relationships. The number of free parameters in the underlying
problem can often be related to the dimension of sets in the
corresponding mathematical model. Since real-life problems usually
depend on very many parameters, the high-dimensional setting is of
particular interest. This is exactly the domain of asymptotic
geometric analysis, which studies quantitative properties of convex
sets (or other geometric structures) as the dimension goes to
infinity. For the last two decades or so the asymptotic theory has
been quite successful in identifying and exploiting "approximate"
symmetries of various problems that escaped the earlier "too
qualitative" or "too rigid" methods of classical functional analysis
and classical geometry. (This led, among others, to the discovery of
many links to computer science.) Finally, to explain our emphasis on
non-commutativity we point out that it simply reflects the fact that
the final outcome of a process may depend on the order of operations
involved; the best known, but by far not the only manifestation of
that principle is quantum mechanics. |
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| NSF Org: |
DMS - Division of Mathematical Sciences |
| Award Number: |
0503642 |
| Award Instrument: |
Continuing grant |
| Program Manager: |
Joe W. Jenkins
DMS Division of Mathematical Sciences
MPS Directorate for Mathematical & Physical Sciences
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| NSF Program(s): |
ANALYSIS PROGRAM |
| Field Application(s): |
Other nsf.applications NEC |
| Program Reference Code(s): |
UNASSIGNED, 0000 |
| Program Element Code(s): |
1281 |
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