Award Abstract # 1100929
Affine algebra representations and discrete integrability

NSF Org: DMS
Division Of Mathematical Sciences
Recipient: UNIVERSITY OF ILLINOIS
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: FY 2011 = $144,321.00
History of Investigator:
  • Rinat Kedem (Principal Investigator)
    rinat@illinois.edu
Recipient Sponsored Research Office: University of Illinois at Urbana-Champaign
506 S WRIGHT ST
URBANA
IL  US  61801-3620
(217)333-2187
Sponsor Congressional District: 13
Primary Place of Performance: University of Illinois at Urbana-Champaign
506 S WRIGHT ST
URBANA
IL  US  61801-3620
Primary Place of Performance
Congressional District:
13
Unique Entity Identifier (UEI): Y8CWNJRCNN91
Parent UEI: V2PHZ2CSCH63
NSF Program(s): ALGEBRA,NUMBER THEORY,AND COM
Primary Program Source: 01001112DB NSF RESEARCH & RELATED ACTIVIT
Program Reference Code(s):
Program Element Code(s): 126400
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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Di Francesco, Kedem "Quantum cluster algebras and fusion products" International Mathematics Research Notices , v.2013 , 2013 , p.50
Di Francesco, Kedem "The solution of quantum A1 T-systems for arbitrary boundary conditions" Communications in mathematical physics , v.313 , 2012 , p.329-350
Di Francesco, Kedem "T-systems with boundaries from network solutions" Electronic Journal of Combinatorics , v.20 , 2013 , p.3
Di Francesco, P; Kedem, R "Non-commutative integrability, paths and quasi-determinants" ADVANCES IN MATHEMATICS , v.228 , 2011 , p.97 View record at Web of Science 10.1016/j.aim.2011.05.01
Philippe Di Francesco and Rinat Kedem "Quantum cluster algebras and fusion products" IMRN , v.2013 , 2013 doi: 10.1093/im4n/rnt004
Philippe Di Francesco and Rinat Kedem "The solution of the quantum A_1 T-system for arbitrary oundary conditions." Communications in Mathematical Physics , v.313 , 2012 , p.329
Philippe Di Francesco and Rinat Kedem "T-systems with boundaries from Network Solutions" The electronic Journal of Combinatorics , v.20 , 2013 , p.#P3
Philippe Di Francesco, Rinat Kedem "Quantum cluster algebras and fusion products" International Mathematics Research Notices , v.2014 , 2014 , p.2593 10.1093/imrn/rnt004
Rinat Kedem, Panupong Vichitkunakorn "T-systems and the pentagram map" Journal of Geometry and Physics , v.87 , 2015 , p.233

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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