| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | May 11, 2018 |
| Latest Amendment Date: | August 4, 2020 |
| Award Number: | 1811111 |
| Award Instrument: | Continuing 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: | September 1, 2018 |
| End Date: | August 31, 2023 (Estimated) |
| Total Intended Award Amount: | $219,017.00 |
| Total Awarded Amount to Date: | $219,017.00 |
| Funds Obligated to Date: |
FY 2019 = $72,997.00 FY 2020 = $74,263.00 |
| History of Investigator: |
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| Recipient Sponsored Research Office: |
415 SOUTH ST WALTHAM MA US 02453-2728 (781)736-2121 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
415 South St Mailstop 116 Waltham MA US 02453-2728 |
| 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: |
01001920DB NSF RESEARCH & RELATED ACTIVIT 01002021DB NSF RESEARCH & RELATED ACTIVIT |
| Program Reference Code(s): | |
| Program Element Code(s): |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
The understanding of the structure of the four-dimensional universe in which we live is a key topic of investigation in modern mathematics and physics. Many of the questions posed by geometers and topologists have to do with the nature of three-dimensional spaces sitting in a four-dimensional space, and with the restrictions that the global properties of larger space places on such sub-spaces. The research in this National Science Foundation funded project uses modern tools of analysis and geometry to shed light on the local nature of such subspaces, including new methods for showing that singularities in such spaces cannot be smoothed. Related analytical techniques will be used to explore the global topology of four-dimensional spaces, including an investigation of their symmetries.
Daniel Ruberman will carry out research in geometric topology, using Seiberg-Witten gauge theory, Heegaard-Floer homology, and more traditional topological techniques. The first parts of the project, joint with Jianfeng Lin and Nikolai Saveliev, are concerned with the smooth topology of four-manifolds that homologically resemble a product of a three-dimensional manifold with a circle. Central questions concern the interpretation of the classical Rohlin invariant and multi-signature invariants in terms of gauge theory; solutions of the main problems will decide the existence of manifolds predicted by high dimensional surgery theory. The PI will work with Adam Levine on embeddings of punctured three-manifolds in four-space, using refined techniques from Heegaard Floer theory to find obstructions. An ongoing project with David Auckly, Hee Jung Kim, and Paul Melvin is concerned with the topology of the diffeomorphism group of a four-dimensional manifold and how it is affected by stabilization of the manifold. Finally, the PI will work with Saveliev and Demetre Kazaras on a new technique to obstruct the existence of positive scalar curvature cobordisms between even-dimensional manifolds.
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.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
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PROJECT OUTCOMES REPORT
Disclaimer
This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.
Intellectual Merit: The PI studied a number of related issues in low dimensional topology, using tools of analysis and geometry, especially those of gauge theory. Low dimensional topology is concerned with global properties of spaces carrying three and four dimensions, which are of great interest in physics as those are the dimensions of our world.
The PI studied analytical, topological, and gauge-theoretic aspects of manifolds with the rational homology of a product of a circle and 3-manifold, which have not been accessible to traditional gauge theory. The results give insight into knot concordance, homology cobordism of 3-manifolds, and spaces of positive curvature metrics. The PI also studied knotted tori in such manifolds, finding a topological counterpart of a gauge-theoretic invariant. A project to study homotopy properties of diffeomorphism groups of 4-manifold was initiated and so far has shown dramatic differences between the corresponding properties of their homeomorphism groups.
Broader Impacts: The research of graduate students at Brandeis University was supervised by the PI, leading to advances in mathematical knowledge. The research of the PI and students supports the overall effort to further the use of advanced mathematical techniques in understanding the geometry of the physical world. The PI helped in disseminating current research in gauge theory through an expository article published in
Last Modified: 12/23/2023
Modified by: Daniel Ruberman
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