Combinatorics of Interacting Particles and Applications

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My research program explores structures that lie at the intersection of combinatorics, representation theory, statistical physics, and integrable systems. The overarching goal is to explore the connections between certain lattice models and orthogonal polynomials through the combinatorial objects that interpolate between them. Relevance. At the center of this proposal is the multi-species asymmetric simple exclusion process (ASEP), which is a 1D exactly solvable stat mech model that is considered a paradigmatic example of non equilibrium systems. The ASEP has applications to traffic flow and translation in protein synthesis. The ASEP also has rich combinatorial structure and deep connections to orthogonal polynomials such as Askey-Wilson, Macdonald, and Koornwinder. These remarkable polynomials play a crucial role in representation theory, algebraic combinatoric, and algebraic geometry, and have been widely studied, though as yet a great many important open questions remain. Background. Macdonald polynomials (of type A) are a two-parameter family that forms a basis for the ring of symmetric functions. A vast body of work has been devoted to understanding their combinatorics due to their inherently complicated nature. The link between statistical mechanics and Macdonald polynomials has been explored in several contexts over the past decade. In particular, it was found that a specialization of the Macdonald polynomial is proportional to the partition function of the multi-species ASEP on a circle. In recent work, with Corteel and Williams formulas were found for these objects in terms of multiline queues. It was found recently that Koornwinder polynomials (Macdonald of type BC root system) are deeply connected to the probabilities of the multi-species ASEP with open boundaries. Extremely little progress has been made on obtaining formulas for probabilities in the multi-species case. In past work, I and my collaborators discovered formulas in the two-species case, thus solving questions that had been open for over ten years. The connections of Macdonald and Koornwinder polynomials to the ASEP and other lattice models has motivated the objectives of this research program. Objectives. The main goals include: Objective 1. Studying the modified Macdonald polynomials using multiline queues [C2], the recently discovered quasisymmetric Macdonald polynomials [C1], and a newly discovered connection to the TAZRP. Objective 2. Studying the combinatorics of the multi-species ASEP with open boundaries and Koornwinder polynomials in order to (1) obtain combinatorial formulas for probabilities of this ASEP, and (2) discover formulas for Koornwinder polynomials via the ASEP.  The success of these objectives will result in major advancement in the understanding of the remarkable Macdonald polynomials and will furthermore have significant impact on the study of lattice models in the field of integrable systems. Note: [C?] refers to publications in my CCV