
NSF Org: |
DMS Division Of Mathematical Sciences |
Recipient: |
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Initial Amendment Date: | May 3, 2011 |
Latest Amendment Date: | May 3, 2011 |
Award Number: | 1100929 |
Award Instrument: | Standard 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: | June 1, 2011 |
End Date: | May 31, 2015 (Estimated) |
Total Intended Award Amount: | $144,321.00 |
Total Awarded Amount to Date: | $144,321.00 |
Funds Obligated to Date: |
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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: |
506 S WRIGHT ST URBANA IL US 61801-3620 |
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: |
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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 research project exploits relations among (1) discrete integrable models in statistical mechanics, (2) representation theory of affine and quantum affine algebras, and (3) the combinatorics of dynamical systems such as cluster algebras. From the physical point of view, the project has applications to wall crossing formulas in string theory and the wave functions of quasi-particles responsible for the quantum Hall effect in condensed matter physics. The problems investigated include fermionic constructions of affine Lie algebra modules and their fusion products; Applications of methods from statistical physics to give explicit solutions for the cluster variables in cluster algebras related to discrete integrable systems such as T-systems or Q-systems, thereby giving proofs of the relevant positivity conjectures; and non-commutative generalizations of these systems, related to the Kontsevich non-commutative wall-crossing formula, or the quantum cluster algebras which describe quantum discrete Liouville or Hirota equations.
This research is at the boundary between mathematics and physics. It seeks to apply techniques from statistical mechanics to solving problems in combinatorics and representation theory of affine Lie algebras and their quantization. The prime characteristic of these problems is integrability, that is, they arise from systems with a high degree of symmetry. Often this symmetry allows for finding explicit solutions in the form of physical partition functions, which are manifestly positive sums over configurations of a system. This positivity property is frequently a conjectured property of the underlying mathematical objects.
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 project uses combinatorial and statistical mechanical methods to find physically useful ways of writing the (characters) graded dimension formulas for spaces which describe quantum field theories, as well as exactly solvable quantum spin chains. The methods come from mathematical physics, whereas the results are directly applicable to the representation theory of affine algebras, quantum affine algebras, and other infinite-dimensional algebras. In addition, some questions in the relatively new theory in algebraic combinatorics, called ``cluster algebras", can be addressed via these models. There are several types of discrete models used in this work. The unifying set of equations are discrete evolution equations called T-systems and their restriction, called Q-systems. These can be regarded as a type of discrete Hirota equations, which are known to be integrable. This means that there is an infinite number of conserved quantities, independent of the discrete time parameter. These conservation laws can be useful in finding the solution, or the solution can be found by other combinatorial or algebraic method. For example, the theory of networks comes into play in the solution of the Q-systems. A network is defined with weights, and the partition function of paths (essentially, current flow) through the network, gives the solutions of the Q-system in terms of given initial data. The same is true for T-systems.
Six publications resulted directly from work during the funding period of this project. The PI gave 18 invited lectures in international conferences, including in Japan, France, England, Scotland, Italy, Canada and South Korea. Honors included a prestigious invitation to give a plenary lecture at the conference on Formal Power Series and Combinatorics in Nagoya, Japan, and an invited lecture at the Internation Congress of Mathematicians in Seoul, South Korea in 2014.
Broader impact:
Two articles by the PI's students, with sole authorship, are in preprint form at the culmination of the funding period. During the funding period, PI gave two summer school lecture series for graduate students, as well as several expository talks for beginning and advanced graduate students. The PI organized one conference, as well as an ongoing seminar for graduate students and several of the professors and postdocs in the area of representation theory and mathematical physics. The PI is the faculty sponsor for the University of Illinois American Women in Mathematics student chapter, a a group which is very active in outreach activities, including organization of an annual mathematics day for high school women in mathematics at the University of Illinois. Currently the PI is mentoring 3 students, two of which are women. During the funding period, the PI also mentored several students who had other advisors, via reading courses and advising, most of whom were female students.
The project interpolates between mathematical physics, notably exactly solvable models in statistical mechanics and conformal field theory, and mathematical constructions in combinatorial representation theory.
Last Modified: 06/12/2015
Modified by: Rinat Kedem
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