| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | June 8, 2005 |
| Latest Amendment Date: | June 8, 2005 |
| Award Number: | 0505740 |
| Award Instrument: | Standard Grant |
| Program Manager: |
Tomek Bartoszynski
tbartosz@nsf.gov (703)292-4885 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | July 1, 2005 |
| End Date: | June 30, 2008 (Estimated) |
| Total Intended Award Amount: | $101,198.00 |
| Total Awarded Amount to Date: | $101,198.00 |
| Funds Obligated to Date: |
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| History of Investigator: |
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| Recipient Sponsored Research Office: |
450 JANE STANFORD WAY STANFORD CA US 94305-2004 (650)723-2300 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
450 JANE STANFORD WAY STANFORD CA US 94305-2004 |
| Primary Place of
Performance Congressional District: |
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| Unique Entity Identifier (UEI): |
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| Parent UEI: |
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| NSF Program(s): | TOPOLOGY |
| Primary Program Source: |
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| Program Reference Code(s): |
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| Program Element Code(s): |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
The proposed activity will continue the development of a relatively new
area of mathematics in which algebraic topology is used to study moduli
spaces of Riemann surfaces and related objects. The main result so far in
this area is the solution, by Madsen and Weiss, of a generalization of a
conjecture of Mumford. This subject lies in the overlap between algebraic
topology and other branches of mathematics, with expected applications in
algebraic geometry, symplectic geometry, and possibly theoretical physics.
The proposed activity will further investigate and develop the result of
Madsen and Weiss. Specifically, the proposed activity will define a
d-dimensional cobordism category C^d and determine the homotopy type of
its classifying space. This will imply the result of Madsen and Weiss,
but is a much more general result, opening for many new possible
applications. At the same time, it will be conceptually simpler and
allows for a much simpler proof. This is one project of the proposal.
The remaining four proposed projects are related.
The theorem of Madsen and Weiss is a very important theorem, that has
already recieved much interest. It sheds a completely new light on the
moduli spaces M_g of Riemann surfaces of genus g and on the 2-dimensional
cobordism category. These objects has been studied and used for long
time, and in many different areas of mathematics. On the other hand it is
clear that the result is not definitive, for example it is restricted to
dimension d=2. The development of the proper generalization should be of
great interest and importance for algebraic topology as well as for
related fields.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
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