| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | January 27, 2017 |
| Latest Amendment Date: | April 5, 2022 |
| Award Number: | 1654159 |
| Award Instrument: | Continuing Grant |
| Program Manager: |
Marian Bocea
mbocea@nsf.gov (703)292-2595 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | September 1, 2017 |
| End Date: | August 31, 2023 (Estimated) |
| Total Intended Award Amount: | $420,000.00 |
| Total Awarded Amount to Date: | $503,540.00 |
| Funds Obligated to Date: |
FY 2018 = $104,818.00 FY 2019 = $121,393.00 FY 2020 = $105,380.00 FY 2021 = $98,482.00 |
| History of Investigator: |
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| Recipient Sponsored Research Office: |
1960 KENNY RD COLUMBUS OH US 43210-1016 (614)688-8735 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
Columbus OH US 43210-1016 |
| 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): |
ANALYSIS PROGRAM, Division Co-Funding: CAREER |
| Primary Program Source: |
01001819DB NSF RESEARCH & RELATED ACTIVIT 01001718DB NSF RESEARCH & RELATED ACTIVIT 01001920DB NSF RESEARCH & RELATED ACTIVIT 01002021DB NSF RESEARCH & RELATED ACTIVIT |
| 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
Symmetry plays an important role in mathematics and science. Classically, the symmetries of a mathematical object form a "group", which is a set with a binary operation such as the integers with addition. In recent decades, we have seen the emergence of quantum mathematical objects whose symmetries form a group-like structure called a "tensor category", which has a collection of objects with a binary fusion operation. Tensor categories are said to encode quantum symmetry: they describe topological phases of matter in physics, and they give us quantum invariants of knots and 3-dimensional surfaces. We are currently seeing the emergence of new mathematical objects which encode "enriched" quantum symmetry, which describe interfaces between 3-dimensional and 2-dimensional quantum systems. At this time, we have several competing formalisms. This project aims to unify these notions and produce exotic examples through classification. The educational component of this project includes undergraduate research and Summer schools on subfactors and quantum symmetry at the Ohio State University. The project will incorporate the principal investigators current learning materials and those developed for these programs into a book on subfactor theory. He will also collaborate with the STEAM Factory at Ohio State University to enhance general scientific and mathematical literacy in the community.
This project has two main focuses: the representation and the classification of these new enriched quantum symmetries. Unitary fusion categories have objects whose dimensions are not necessarily integers, so representing unitary fusion categories requires von Neumann factors, whose modules have a notion of continuous dimension. In this project the principal investigator will use his previous experience in the classification of small index subfactors to classify quantum symmetries enriched in small unitary ribbon categories. This will study an enriched operator algebra theory to develop an enriched subfactor theory. The principal investigator will also develop the theory of bicommutant categories, which are a higher categorical analog of von Neumann algebras originally due to Henriques. These bicommutant categories have important connections to conformal field theory, and they are expected to be an important tool in the classification of enriched quantum symmetries.
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.
Classically, the notion of symmetry in mathematics and the sciences is encoded by the notion of a group. A group of symmetries acts on an object as a collection of self-maps. In recent decades, we have seen that quantum mathematical objects live in higher categories, and thus they admit higher quantum symmetries in the form of tensor categories. Von Neumann algebras are examples of such quantum objects, which can be viewed as either non-commutative probability spaces, or as local algebras of observables in quantum systems. Tensor categories act on von Neumann algebras via their bimodules.
The intellectual merit of this award focused on two main research thrusts: anchored planar algebras and bicommutant categories. Anchored planar algebras provide a graphical calculus to study enriched quantum symmetries, which arise in many areas of mathematics and physics, most recently in topological order in theoretical condensed matter physics. The PI developed the theory of anchored planar algebras, proving a complete characterization in terms of module tensor categories for the enriching theory. In one notable result, the PI used anchored planar algebra technology and enriched fusion categories to study composition of domain walls between (2+1)D topological orders. He also studied unitarity for enriched fusion categories and anchored planar algebras, which led to an investigation of unitarity for higher dagger categories. Unitarity is often overlooked in the theory of tensor categories, but it is an important ingredient for physical systems.
A bicommutant category is a categorification of the notion of a von Neumann algebra. The PI studied examples of commutant categories coming from representations of fusion categories as bimodules over von Neumann algebras. A recent joint article of the PI is the culmination of the program of studying anchored planar algebras together with bicommutant categories, giving a classification of certain finite objects related to these commutant categories in terms of an anchored planar algebra as an invariant. This result can be viewed as extending the classification of finite depth bimodules of the hyperfinite II_1 factor by Jones' planar algebras in the theory of subfactors.
The broader impacts of this award include training of graduate students, mentoring of undergraduate research, talks for the general public, and a summer workshop in 2019 on quantum symmetries. Over the 6 years of this award, the PI mentored 13 undergraduate researchers and graduated 4 PhD students and is currently mentoring another 6 PhD students.
Last Modified: 01/04/2024
Modified by: David S Penneys
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