| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | August 6, 2013 |
| Latest Amendment Date: | August 6, 2013 |
| Award Number: | 1311911 |
| Award Instrument: | Standard Grant |
| Program Manager: |
Joanna Kania-Bartoszynska
jkaniaba@nsf.gov (703)292-4881 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | August 15, 2013 |
| End Date: | July 31, 2017 (Estimated) |
| Total Intended Award Amount: | $188,871.00 |
| Total Awarded Amount to Date: | $188,871.00 |
| Funds Obligated to Date: |
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| History of Investigator: |
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| Recipient Sponsored Research Office: |
202 HIMES HALL BATON ROUGE LA US 70803-0001 (225)578-2760 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
LA US 70803-2701 |
| 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
Gilmer will continue to study integral topological quantum field theories (TQFT) and their applications to low dimensional topology. He will continue to study representations of mapping class groups that can be defined using integral topological quantum field theories. In particular, he will study the induced modular representations. Gilmer plans to compute strong shift equivalence class invariants of knots and other spaces, which can be defined using TQFT. He wishes to calculate further strong shift equivalence invariants to uncover their topological meaning. He has used topological quantum field theory invariants to find obstructions to fibered knots being ribbon knots. He wants to see if there are further obstructions. In general, he will try to use quantum topology as a tool in low-dimensional topology. Gilmer also plans to use 4-dimensional topology to study the topology of real algebraic curves in the real projective plane.
Topology is the study of intrinsic shape. It is sometimes called "rubber sheet" geometry as the objects that one studies can be twisted and stretched but not torn. Recently topology has experienced a large influx of ideas from physics. Topological quantum field theory is one of the most current and exciting areas of topology with intimate connections to high energy physics, quantum computing as well as other areas of mathematics, for instance number theory. Gilmer will use topological quantum field theory as a tool to study low dimensional topology. Low dimensional topology is important for chemistry and biology as it has implications for the mechanism of DNA, and other molecular configurations.
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PROJECT OUTCOMES REPORT
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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.
Topology is the study of generalized shapes considered up to deformations which preserve the relation of nearness. Sometimes topology is called ``rubber sheet geometry’’ as twisting and bending is allowed but tearing is not allowed. Low dimensional topology concerns certain shapes (called manifolds) which have dimension four or less. Topology has many special features in these dimensions. We studied 3-dimensional manifolds, one dimensional manifolds in 3-dimensional manifolds (links), and symmetries of 2-dimensional manifolds (mapping classes of surfaces). We studied numerical invariants of links and found new ways to relate these invariants to each other, and new ways to calculate these invariants. We used link invariants to study how the set of the zeros of a polynomial in two variables can look. This was one application of our work in topology to the general area known as algebra. We were also able to answer questions in another area of algebra (called representation theory) which is concerned with algebraic symmetries. One of the tools we used goes by the name topological quantum field theory (abbreviated TQFT). The idea for TQFT came from physics. TQFT has had many beautiful applications to low dimensional topology.
I mentored one post-doc. I directed the Ph.D. dissertation research of two graduate students. I introduced TQFT to four beginning graduate students during summer reading courses.
Last Modified: 08/03/2017
Modified by: Patrick M Gilmer
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