Research

My primary research interests lie in algebraic combinatorics and discrete geometry, including symmetric functions, polytopes, matroids, combinatorial representation theory, and Schubert calculus.

Code that I have written in Mathematica for my research is available on my Github page.

Preprints and Publications

We use a new tool called bubbling diagrams to study the support of vexillary Grothendieck polynomials. We show their top-degree component has support a Schubitope.


We prove a minimality conjecture of Reiner, Woo, and Yong regarding the presentation of the cohomology ring of any Schubert variety.


Using climbing chains, we identify the leading terms of each homogeneous component of any Grothendieck polynomial. Consequently, we give a new proof of Pechenik, Speyer, and Weigandt's degree formula for Grothendieck polynomials.


We analyze the combinatorial structure of the support of Grothendieck polynomials, Taking the new perspective of the support as a poset, we make various conjectures and prove special cases.


We study the principal specialization (setting all variables to 1) of dual characters of flagged Weyl modules, a generalization of both Schubert and key polynomials.


We give a new operator formula for Grothendieck polynomials, generalizing Magyar's Demazure operator formula for Schubert polynomials. We apply our formula to prove a necessary divisibility condition for a monomial to appear in a given Grothendieck polynomial. 


We connect generalized permutahedra with Schubert calculus, giving a (polynomial-time) sufficient vanishing criteria for Schubert intersection numbers of the flag variety.


We give explicit formulas for the degree of the Grothendieck polynomial of a Grassmannian permutation. We apply our formulas to give an expression for the Castelnuovo-Mumford regularity of the Schubert determinantal ideal of a Grassmannian permutation.


We show that normalized Schur polynomials are strongly log-concave functions. Consequently, we prove a special case of Okounkov's conjecture for Littlewood–Richardson coefficients.


We characterize the permutations whose Schubert polynomials have all coefficients lying in {0,1} by a set of twelve avoided patterns. We give a term-by-term embedding of one Schubert polynomial into another whenever pattern containment occurs.


We derive consequences of the fact that Gelfand-Tsetlin polytopes are both flow polytopes and marked order polytopes. We then establish a general theorem connecting marked order polytopes and flow polytopes. 


For any permutation $w \in S_n$ with column-convex Rothe diagram, we construct a polytope $P_w$ whose integer point transform projects to the Schubert polynomial of $w$. We show that $P_w$ is a Minkowski sum of Gelfand-Tsetlin polytopes.


We show that the dual character of the flagged Weyl module of any diagram has saturated Newton polytope which is a generalized permutahedron. As  special cases, these results hold for Schubert and key polynomials.


We study a polynomial invariant of a family of subdivisions of certain flow polytopes. We show that the invariant has saturated Newton polytope and that the Newton polytope of each homogeneous component is a generalized permutahedron. We deduce the same holds for a family of Grothendieck polynomials.