| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | September 6, 2000 |
| Latest Amendment Date: | August 1, 2002 |
| Award Number: | 0071520 |
| Award Instrument: | Standard Grant |
| Program Manager: |
Junping Wang
jwang@nsf.gov (703)292-4488 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | September 15, 2000 |
| End Date: | August 31, 2004 (Estimated) |
| Total Intended Award Amount: | $97,531.00 |
| Total Awarded Amount to Date: | $105,564.00 |
| Funds Obligated to Date: |
FY 2002 = $8,033.00 |
| History of Investigator: |
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| Recipient Sponsored Research Office: |
506 S WRIGHT ST URBANA IL US 61801-3620 (217)333-2187 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
506 S WRIGHT ST URBANA IL US 61801-3620 |
| 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): |
GEOMETRIC ANALYSIS, COMPUTATIONAL MATHEMATICS, NUMERIC, SYMBOLIC & GEO COMPUT |
| Primary Program Source: |
app-0102 |
| 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
Sullivan
0071520
The investigator, with his collaborators, studies geometric optimization problems like finding minimum-energy shapes for surfaces and knots in space. They extend their recent classification of embedded constant-mean-curvature surfaces with three ends to the more general case of surfaces with any number of coplanar ends, and also investigate in detail surfaces with truncated ends. In addition, the investigator computes these surfaces numerically, in order, for instance, to create interactive computer graphics. This project uses Willmore's elastic bending energy, and its gradient flow, to discover new minimal surfaces in euclidean and spherical space. The Willmore flow has been recently shown to have short-time solutions, but the investigator considers whether it can fail to have long-time solutions. This project also studies configurations for knots which minimize ropelength, giving new lower bounds for the ropelength of small knots, and new asymptotic bounds on the growth of ropelength with crossing number. Finally, the investigator uses his experience with numerical modeling of curves and surfaces to give new understanding of geometrically natural discretizations for quantities related to curvature.
Many real-world problems can be cast in the form of optimizing some feature of a shape; mathematically, these become variational problems for geometric energies. For instance, thin films, like those in foams, usually minimize their area and thus are constant-mean curvature surfaces. Cell membranes are more complicated bilayer surfaces which minimize an elastic bending energy known mathematically as the Willmore energy. Knotted curves achieve an optimal shape when a rope is pulled tight, or if a charged knotted wire repels itself electrostatically; understanding such configurations helps explain the behavior of biological molecules like DNA. This project explores such phsically natural problems, which remain challenging from both theoretical and computational standpoints.
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