
NSF Org: |
DMS Division Of Mathematical Sciences |
Recipient: |
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Initial Amendment Date: | April 18, 2018 |
Latest Amendment Date: | April 13, 2020 |
Award Number: | 1802044 |
Award Instrument: | Continuing Grant |
Program Manager: |
James Matthew Douglass
mdouglas@nsf.gov (703)292-2467 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
Start Date: | May 1, 2018 |
End Date: | April 30, 2023 (Estimated) |
Total Intended Award Amount: | $265,000.00 |
Total Awarded Amount to Date: | $265,000.00 |
Funds Obligated to Date: |
FY 2019 = $88,334.00 FY 2020 = $88,332.00 |
History of Investigator: |
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Recipient Sponsored Research Office: |
506 S WRIGHT ST URBANA IL US 61801-3620 (217)333-2187 |
Sponsor Congressional District: |
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Primary Place of Performance: |
Urbana IL US 61801-2943 |
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): | ALGEBRA,NUMBER THEORY,AND COM |
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
This research project investigates questions at the intersection of mathematics and theoretical physics. The project is focused on questions that originate in statistical mechanics, which describes the average behavior of systems with large numbers of components. The algebraic study of these systems, and their symmetries, leads to new algebras, whose structure is in turn encoded combinatorially by collections of functions that arise as solutions to systems of equations. These new algebras, their generalizations, and the resulting combinatorics are the main topics to be studied. The research will be conducted in association with students at the graduate and undergraduate level, and will enhance our understanding of how these fields of study connect to each other.
More precisely, the project concerns graded tensor products of cyclic current algebra modules, specifically Kirillov-Reshetikhin modules. The combinatorial framework is the quantum cluster algebra of the associated Q-systems, the solutions of which can be expressed in terms of plane tilings and non-intersecting path models. By investigating the discrete integrability of these systems, the investigators expect to produce conserved quantities, so far achieved only in type A. In turn, these should lead to difference equations for the graded characters of tensor product product, which generalize the difference Toda equations. Moreover, the solutions of these equations can be expressed as iterated action of difference creation operators, closely related to generalized Macdonald operators defined within the context of polynomial representations of double affine Hecke algebras. Relations to quantum affine and elliptic Hall algebras will also be investigated, as well as more combinatorial questions such as the existence of some (quantum) non-commutative counterpart to the Gessel-Viennot determinant for non-intersecting path enumeration.
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.
The property of integrability in physical systems gives rise to interesting algebraic, geometric and combinatorial structures. Integrability in a system means that the system has sufficient symmetries, or equivalently conserved quantities, so that it can be solved exactly. In this project, we study families of discrete integrable systems which originated in two-dimensional lattice models or one-dimensional quantum spin chains. In previous decades, these systems have led to the discovery of deformations of Lie algebras known as quantum groups, which are related to topological invariant theory. In this project, their relation to the more recent algebraic-combinatorial subject of cluster algebras is explored, as well as Macdonald theory and the theory of finite difference operators.
The project consisted of two main directions. In the first, the full family of discrete evolutions known as Q-systems and their quantizations was studied. The relation to Macdonald theory and Macdonald-Koornwinder difference operators was explained. The second part concentrated on the asymptotic properties of solutions of T-systems, a discrete evolution which gives rise to Q-systems, subject to different types of boundary conditions. Solutions to these systems may be interpreted in terms of statistical models of dimers/tilings which display a property known as the arctic phenomenon, namely the emergence of a sharp phase separation between crystalline and liquid regions, for large systems. More generally we also explored the arctic phenomenon for interacting vertex models (see attached figure).
The project involved the training and mentorship of graduate students and a postdoctoral fellow at the University of Illinois. One student completed his PhD during the period of the grant and two more are nearing completion. A graduate student learning seminar on cluster algebras and integrable systems was started during the period of the grant and included several additional graduate students. Advanced graduate courses were offered at the University of Illinois for students in mathematics and physics.
Last Modified: 05/04/2023
Modified by: Philippe Di Francesco
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