| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | April 28, 2022 |
| Latest Amendment Date: | April 28, 2022 |
| Award Number: | 2203498 |
| Award Instrument: | Standard Grant |
| Program Manager: |
Swatee Naik
snaik@nsf.gov (703)292-4876 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | July 1, 2022 |
| End Date: | October 31, 2025 (Estimated) |
| Total Intended Award Amount: | $282,177.00 |
| Total Awarded Amount to Date: | $282,177.00 |
| Funds Obligated to Date: |
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| History of Investigator: |
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| Recipient Sponsored Research Office: |
615 W 131ST ST NEW YORK NY US 10027-7922 (212)854-6851 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
2990 Broadway, Rm 613 MC 4419 New York NY US 10027-6902 |
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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 Element Code(s): |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
The goal of topology is to identify which features of a shape do not change under a continuous deformation, with concrete applications in many areas of science such as condensed matter physics, cosmology, data analysis, and biology. As one can infer information about the shape of a drum by listening to the way it sounds, one can define topological invariants of spaces of dimension 3 and 4 by studying the solutions of certain partial differential equations naturally arising in gauge theory, the geometric language in which the fundamental laws of the Standard Model of particle physics are formulated. This project focuses on the Seiberg-Witten equations which, because of their geometric nature, provide a perfect vantage point to probe the interactions of topology with neighboring subjects such as hyperbolic geometry, spectral theory, and complex analysis. A key objective is the exploration of new avenues of investigation at the interface with these fields of mathematics. Towards this end, this project will also create many research opportunities both at the undergraduate and graduate level.
The PI will study monopole Floer homology, with the following main goals: explore the interactions with hyperbolic geometry in three dimensions using tools from spectral geometry, the theory of elliptic partial differential equations, and the Selberg trace formula; develop computational tools for the maps induced by general negative definite cobordisms using index theory and functoriality properties of coupled Morse homology; investigate possible relations with classical topics in algebraic geometry such as the study of the singularities of the theta divisor of a Riemann surface; and use Pin(2)-symmetry to understand potential geometric characterizations of rational homology spheres with small Floer homology, in the spirit of the L-space conjecture.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
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