How we can use AI in Math work - talk slides
Grothendieck Shenanigans
The story started with a problem of Stanley from his 2017 paper “Some Schubert shenanigans”: what is the asymptotically maximal value of a principal specialization of a Schubert polynomial S_w, and what does the corresponding permutation w look like? While this question is still out of reach, we investigate its variant for nonsymmetric Grothendieck polynomials. We analyze the asymptotic behavior of random permutations whose probabilities are proportional to the Grothendieck polynomials using TASEP and tools from integrable probability. The typical behavior of these permutations is described by a specific permuton (a continuous analog of a permutation). We also examine a family of atypical layered permutations that achieve the same asymptotically maximal weight.
Based on the joint work with A.H. Morales, G. Panova, and D. Yeliussizov.
Hook-length Formulas for Skew Shapes via Contour Integrals and Vertex Models
The number of standard Young tableaux of a skew shape $\lambda/\mu$ can be computed as a sum over excited diagrams inside $\lambda$. Excited diagrams are in bijection with certain lozenge tilings, with flagged semistandard tableaux and also nonintersecting lattice paths inside $\lambda$. We give two new proofs of a multivariate generalization of this formula, which allow us to extend the setup beyond standard Young tableaux and the underlying Schur symmetric polynomials. The first proof uses multiple contour integrals. The second one interprets excited diagrams as configurations of a six-vertex model at a free fermion point, and derives the formula for the number of standard Young tableaux of a skew shape from the Yang-Baxter equation.
Blue Ridge Probability Day
MATH 2310 • Calculus III (2 sections)
Grothendieck Shenanigans: Permutons from Pipe Dreams via Integrable Probability
We study random permutations arising from reduced pipe dreams. Our main model is motivated by Grothendieck polynomials with parameter $\beta=1$ arising in K-theory of the flag variety. The probability weight of a permutation is proportional to the principal specialization (setting all variables to 1) of the corresponding Grothendieck polynomial. By mapping this random permutation to a version of TASEP (Totally Asymmetric Simple Exclusion Process), we describe the limiting permuton and fluctuations around it as the order $n$ of the permutation grows to infinity. The fluctuations are of order $n^{\frac{1}{3}}$ and have the Tracy-Widom GUE distribution, which places this algebraic (K-theoretic) model into the Kardar-Parisi-Zhang universality class.
We also investigate non-reduced pipe dreams and make progress on a recent open problem on the asymptotic number of inversions of the resulting permutation. Inspired by Stanley’s question for the maximal value of principal specializations of Schubert polynomials, we resolve the analogous question for $\beta=1$ Grothendieck polynomials, and provide bounds for general $\beta$.