| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | July 6, 2017 |
| Latest Amendment Date: | May 10, 2022 |
| Award Number: | 1664387 |
| 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: | July 1, 2017 |
| End Date: | June 30, 2023 (Estimated) |
| Total Intended Award Amount: | $149,209.00 |
| Total Awarded Amount to Date: | $149,209.00 |
| Funds Obligated to Date: |
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| History of Investigator: |
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| Recipient Sponsored Research Office: |
1000 OLD MAIN HL LOGAN UT US 84322-1000 (435)797-1226 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
UT US 84322-1415 |
| 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
Mathematics inspired by physics has successfully provided the background and the language for most sophisticated areas of modern physics. In this project the principal investigators aim to create new algebraic and geometric tools helping to formulate ideas and methods of quantum physics in a precise mathematical way. Specifically, the team will combine its members' past experience and achievements to construct new Topological Quantum Field Theories for manifolds with additional structures. Inspired by Witten's Chern-Simons theory and invariants of 3-manifolds that are known now as Witten-Reshetikhin-Turaev invariants, the principal investigators will explore their new Field Theories with the objective to find physical definitions for the resulting topological invariants of manifolds. This work is stimulated by several fundamental examples and will open the door to new research avenues in algebra, topology, geometry, mathematical physics, and related areas of mathematics. The broader impacts of the project belong to two main categories: mentoring and outreach. The members of the research team are currently mentoring a total of twelve PhD students. They will advise graduate students on projects related to the main objectives of the grant. The outreach component is to organize several workshops and conferences aimed at developing communication and collaborative research between participants of the project, establishing scientific connections with other mathematicians, as well as fostering broader applications of this work.
In 1999, Turaev introduced Homotopy Quantum Field Theories (HQFTs), which are generalizations of Topological Quantum Field Theories (TQFTs) studied by Schwartz, Witten, and Atiyah. HQFTs produce topological invariants of manifolds furnished with extra data that add supplementary topology/geometry to the context of TQFTs. Most of the theory of quantum invariants and HQFTs involves monoidal categories which have certain additional properties like being semi-simple. In various collaborations started in 2005, the team members developed a theory of re-normalized Quantum Invariants that derives non-trivial topological invariants from non-semi-simple categories. In this project the principal investigators will further use renormalization to develop non-trivial HQFTs, based on the examples coming from the theory of unrestricted quantum group. These new HQFTs should share the strength and the new features of the re-normalized Quantum Invariants. The principal investigators will further search for a physical interpretation of these new invariants.
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.
This grant aimed to create and investigate new renormalized quantum invariants coming from non semisimple categories and Homotopy Quantum Field Theories (HQFTs) of manifolds with additional structure. The grant had three main intellectual merit goals involving : 1) define renormalized quantum invariants of link with flat SL(2,C)-connections in their complements, 2) extend these invariants to 3-manifold invariants and 3) explore the physical theory underlying these invariants. Solutions to the first and third goal have been given in the papers "Holonomy braidings, biquandles and quantum invariants of links with SL(2,C) flat connections" and "A QFT for non-semisimple TQFT," respectively. Progress on the second goal has been made and is given in other papers. The work of this FRG grant has also lead to many new research avenues and techniques, including new ideas to approach the second goal. The main broader impacts were centered around mentoring and outreach. In particular, the team advised the seven PhD students involved in the FRG grant (all graduated in 2020). The team also organized two conferences: "New Developments in Quantum Topology'' at University of California, Berkeley and "Quantum Topology and Geometry" at Institut Henri Poincaré, Paris. These conferences created connections between different research areas and educated junior mathematicians.
Last Modified: 09/25/2023
Modified by: Nathan C Geer
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