Applied Asymptotic Algebraic Combinatorics

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Algebraic combinatorics leverages the power of algebra to analyze discrete structures. It is a powerful approach to answering combinatorial questions, often leading to exact and explicit enumerative formulas that are not obtainable from first principles. Asymptotic algebraic combinatorics deals with discrete structures whose defining parameters are extremely large, far exceeding the numerical range of everyday physical experience. In this setting, exact formulas become unwieldy and unusable; asymptotic algebraic combinatorics leverages algebraic methods to obtain useful approximations that are typically not accessible. The algebraic approach to asymptotic combinatorics is especially pertinent in the age of big data, where discrete structures loom large over the information landscape, but the tools to handle them are in short supply. This research project intends employ algebraic techniques to further develop useful tools in asymptotic combinatorics. The project will involve graduate students in the research.This project aims to develop new algebraic methods for the asymptotic analysis of large structures that appear in probability and mathematical physics. On the probabilistic side, the PI will build a theory of asymptotic Fourier analysis for large random matrices using recent analysis of large rank orbital integrals as a foundation. One of the main goals of this endeavor is to provide a toolbox that can be uniformly applied to both large random matrices and their quantized counterparts, random representations of large Lie groups. On the mathematical physics side, the PI plans to use the algebraic techniques underlying recent analysis of large rank link integrals to undertake a rigorous study of Yang-Mills partition functions, first revisiting the two-dimensional case and then moving to higher dimensions via link integrals. A key goal here is to rigorously understand combinatorics of asymptotic freedom and gauge-string dualities.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.