Research

I am a PhD student in combinatorial algebraic geometry, working in the group of algebraic combinatorists in the Department of Combinatorics and Optimization at the University of Waterloo. I am partially supported by NSERC and a University of Waterloo President's Graduate Scholarship.

My research is in combinatorial and computational algebraic geometry. Recently, I have been thinking about Springer fibers, Hessenberg varieties, and matrix Schubert varieties, and thinking aobut tools such as geometric vertex decomposition and Frobenius splitting.

Papers

My papers are also available on the arXiv. You can also view my profiles on Google Scholar, MathSciNet, and zbMath.

1. Webs and smooth components of two column Springer fibers
   Submitted

   Abstract. Webs and Springer fibers are separately important objects in representation theory: webs give a diagrammatic calculus for tensor invariants of 𝔰𝔩_k, and the cohomology group of Springer fibers can be used to construct the irreducible representations of the symmetric group. Fung's 1997 thesis gave the first evidence of a connection between 𝔰𝔩₂ webs and Springer fibers, showing that webs naturally index and describe the components of certain "two row" Springer fibers. However, this case is known to be far from generic.
   
   This paper deepens this connection with a similar correspondence in the substantially more complicated "two column" case. In particular, and building on works of Fresse, Melnikov, and Sakas-Obeid, we use webs to give a clean characterization of the smooth components of two column rectangle Springer fibers and a simple description of the geometry of these smooth components. We also show that the Poincaré polynomial of the smooth components is invariant under the natural dihedral action on the corresponding webs.

2. Corrigendum to "Counting two-column Young tableaux corresponding to smooth components of Springer fibers" [J. Algebr. Comb. 61:18 (2025)] (original paper by R. Mansour)
   with Ronit Mansour
   To appear, Journal of Algebraic Combinatorics
3. The GeometricDecomposability package for Macaulay2
   with Adam Van Tuyl
   Journal of Software for Algebra and Geometry 14 (2024), no. 1, 41–50

   Abstract. Using the geometric vertex decomposition property first defined by Knutson, Miller, and Yong, a recursive definition for geometrically vertex decomposable ideals was given by Klein and Rajchgot. We introduce the Macaulay2 package GeometricDecomposability which provides a suite of tools to experiment and test the geometric vertex decomposability property of an ideal.

4. Gröbner geometry for regular nilpotent Hessenberg Schubert cells
   with Sergio Da Silva, Megumi Harada, and Jenna Rajchgot
   Journal of Pure and Applied Algebra 228 (2024), no. 7, 107648

   Abstract. A regular nilpotent Hessenberg Schubert cell is the intersection of a regular nilpotent Hessenberg variety with a Schubert cell. In this paper, we describe a set of minimal generators of the defining ideal of a regular nilpotent Hessenberg Schubert cell in the type A setting. We show that these minimal generators are a Gröbner basis for an appropriate lexicographic monomial order. As a consequence, we obtain a new computational-algebraic proof, in type A, of Tymoczko’s result that regular nilpotent Hessenberg varieties are paved by affine spaces. In addition, we prove that these defining ideals are complete intersections, are geometrically vertex decomposable, and compute their Hilbert series. We also produce a Frobenius splitting of each Schubert cell that compatibly splits all of the regular nilpotent Hessenberg Schubert cells contained in it. This work builds on, and extends, work of the second and third author on defining ideals of intersections of regular nilpotent Hessenberg varieties with the (open) Schubert cell associated to the Bruhat-longest permutation.

5. Geometric vertex decomposition and liaison for toric ideals of graphs
   with Sergio Da Silva, Jenna Rajchgot, and Adam Van Tuyl
   Algebraic Combinatorics 6 (2023), no. 4, 965–997
   

   Abstract. Geometric vertex decomposability for polynomial ideals is an ideal-theoretic generalization of vertex decomposability for simplicial complexes. Indeed, a homogeneous geometrically vertex decomposable ideal is radical and Cohen-Macaulay, and is in the Gorenstein liaison class of a complete intersection (glicci).

   In this paper, we initiate an investigation into when the toric ideal I_G of a finite simple graph G is geometrically vertex decomposable. We first show how geometric vertex decomposability behaves under tensor products, which allows us to restrict to connected graphs. We then describe a graph operation that preserves geometric vertex decomposability, thus allowing us to build many graphs whose corresponding toric ideals are geometrically vertex decomposable. Using work of Constantinescu and Gorla, we prove that toric ideals of bipartite graphs are geometrically vertex decomposable. We also propose a conjecture that all toric ideals of graphs with a square-free degeneration with respect to a lexicographic order are geometrically vertex decomposable. As evidence, we prove the conjecture in the case that the universal Gröbner basis of I_G is a set of quadratic binomials. We also prove that some other families of graphs have the property that I_G is glicci.

Software

1. GeometricDecomposability, a Macaulay2 package to check whether ideals are geometrically vertex decomposable, and related methods
   with Adam Van Tuyl
   For changes by version, see the changelog

I also wrote the IntegerProgramming package for Macaulay2 package as part of a project for the course Computational Commutative Algebra and Algebraic Geometry with Mike Stillman at the Fields Institute in Winter 2025.

Theses

1. Gröbner Geometry for Hessenberg Varieties
   Master's Thesis, McMaster University, 2024
   Supervisor: Jenna Rajchgot

   Abstract. We study Hessenberg varieties in type A via their local defining equations, called patch ideals. We focus on two main classes of Hessenberg varieties: those associated to a regular nilpotent operator and those associated to a semisimple operator.

   In the setting of regular semisimple Hessenberg varieties, which are known to be smooth and irreducible, we determine that their patch ideals are triangular complete intersections, as defined by Da Silva and Harada. For semisimple Hessenberg varieties, we give a partial positive answer to a conjecture of Insko and Precup that a given family of set-theoretic local defining ideals are radical.

   A regular nilpotent Hessenberg Schubert cell is the intersection of a Schubert cell with a regular nilpotent Hessenberg variety. Following the work of the author with Da Silva, Harada, and Rajchgot, we construct an embedding of the regular nilpotent Hessenberg Schubert cells into the coordinate chart of the regular nilpotent Hessenberg variety corresponding to the longest-word permutation in Bruhat order. This allows us to use work of Da Silva and Harada to conclude that regular nilpotent Hessenberg Schubert cells are also local triangular complete intersections.

2. Geometric Vertex Decomposition and Hessenberg Patch Ideals
   Undergraduate Thesis, McMaster University, 2022
   Supervisors: Sergio Da Silva, Megumi Harada, and Jenna Rajchgot

Many then-current and former members of McMaster's algebra group attended CAAC 2023 at the University of Waterloo.

Top row: Illya Kierkosz, Jenna Rajchgot, Kieran Bhaskara, myself, Adam Van Tuyl, Adrian Cook.
Bottom row: Thái Thành Nguyễn, Federico Galetto, Megumi Harada, Graham Keiper, Sergio Da Silva, Büşra Atar, Runyue Wang.