Cluster Algebras, the Ising Model, and Affine Kazhdan-Lusztig Cells

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The PI is interested in algebraic combinatorics and its relations to other areas of mathematics. Algebraic combinatorics is a unique area in that it interacts closely with a truly wide range of scientific subjects. For example, cluster algebras studied by the PI arise in many areas of mathematics (representation theory, geometry, combinatorics, dynamical systems) and physics (string theory, scattering amplitudes, integrable systems). Discovered around the year 2000 cluster algebras provide an amazing unifying framework for many questions. The PI is interested in deepening these connections, and finding even more ways in which mathematics and physics are a single science. The projects also provide research training opportunities for graduate students.The four projects the PI will work on pertain to cluster algebras, the Ising model, and Kazhdan-Lusztig theory. In the first project the PI plans to continue his work on classifying periodic and integrable T-systems, as well as studying their properties. In the second project the PI plans to work on a tensor diagram approach to higher Teichmuller theory. In a recent work the PI and a coauthor have used Postnikov's plabic graphs to give a comprehensive description of correlation functions of the Ising model in terms of inequalities. In the third project the PI plans to work on using those results to derive properties of limits of correlation functions, as well as on extending those results to other statistical mechanics models. Finally, in the fourth project the PI plans to use the generalized Robinson-Schensted insertion he developed to study affine Kazhdan-Lusztig theory.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.