Combinatorial State Sums and Interval Flag Varieties

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This research project addresses questions in algebraic combinatorics and algebraic geometry, areas of mathematics concerned with the description of symmetry and with the solution of multivariate polynomial equations. The project involves the study of "state sums" -- a catchall term from statistical mechanics to describe certain sums of products (typically of polynomials) -- and the study of topological quantum field theory, which arose in physics and has turned out to have connections to several areas of modern mathematics. Many important quantities in algebraic combinatorics can be computed as state sums; these are often sums over all the labelings of a quadrangulated surface with compatible tiles, like assembling a jigsaw puzzle. The remarkable fact is that these totals are frequently independent of the quadrangulation, suggesting that they may have some more fundamental definition. This project investigates a source from topological quantum field theory that might unify these state sum results, while suggesting ways to probe them more deeply. Specifically, changing a quadrangulation on a surface relates to evolving it through space, which suggests exploring both higher- and lower-dimensional analogues. This program has many subprojects suitable for graduate students in mathematics and statistical mechanics, who will be involved in the research.This research program comprises two projects. The first project uses inspiration from quantum field theory and geometric representation theory in an endeavor to elucidate polynomial formulae arising in algebraic combinatorics. Formulae for interesting polynomials often take the form of sums of products of linear polynomials. For example, the "Schubert polynomial" associated to a permutation can be written as a sum over certain 2-d diagrams for the permutation. The project will look more deeply into these polynomial formulae, in two steps. The first step is to see a polynomial as a matrix coefficient inside a time-evolution operator in a (1+1)-dimensional quantum field theory. Since long-time-evolution is a composite of many short-time evolutions, the matrix might be expressed as a product of simpler matrices, giving exactly the sum over products. The second step is to regard the Hilbert spaces of these quantum field theories as homology groups of certain algebraic varieties, after the manner of geometric representation theory. This would naturally imply some commutation properties of these matrices, such as the Yang-Baxter equation. The second project aims to develop a generalization of Schubert calculus to chains of linear subspaces.