Research

Below is the list of my papers and preprints, organized chronologically.

1. On minimal symplectic alternating algebras (with Layla Sorkatti and Manisha Varahagiri – 2023) https://doi.org/10.48550/arXiv.2311.11987
   Abstract: In this paper, we study the dual subclass of algebras of minimal class. In particular, we show that symplectic alternating algebras of dimension up to 16 that are minimal, in the sense that they are of rank 2 with minimum nilpotency class, have a class that confirm a conjecture that has been raised in Layla Sorkatti’s previous work.
   
2. K-Orbit closures and Hessenberg varieties (with Mahir Bilen Can, Martha Precup, and John Shareshian – 2023) https://doi.org/10.48550/arXiv.2309.05770
   Abstract: This article explores the relationship between Hessenberg varieties associated with semisimple operators with two eigenvalues and orbit closures of a spherical subgroup of the general linear group. We establish the specific conditions under which these semisimple Hessenberg varieties are irreducible. We determine the dimension of each irreducible Hessenberg variety under consideration and show that the number of such varieties is a Catalan number. We then apply a theorem of Brion to compute a polynomial representative for the cohomology class of each such variety. Additionally, we calculate the intersections of a standard (Schubert) hyperplane section of the flag variety with each of our Hessenberg varieties and prove this intersection possess a cohomological multiplicity-free property.
   
3. DIII clan combinatorics for the orthogonal Grassmannian (with Aram Bingham – 2021)
   Australasian Journal of Combinatorics, Volume 79(1), Pages 55–86.
   Abstract: Borel subgroup orbits of the classical symmetric space SO(2n)/GL(n) are parametrized by DIII (n, n)-clans. We group the clans into “sects” corresponding to Schubert cells of the orthogonal Grassmannian, thus providing a cell decomposition for SO(2n)/GL(n). We also compute a recurrence for the rank polynomial of the weak order poset on DIII clans, and then describe explicit bijections between such clans, diagonally symmetric rook placements, certain pairs of minimally intersecting set partitions, and a class of weighted Delannoy paths. Clans of the largest sect are in bijection with fixed-point-free partial involutions.
   
4. Sects and lattice paths over the Lagrangian Grassmannian (with Aram Bingham – 2020)
   The Electronic Journal of Combinatorics, Volume 27, Issue 1, Paper No. 1.51, 28. https://doi.org/10.37236/8664.
   Abstract: We examine Borel subgroup orbits in the classical symmetric space of type CI, which are parametrized by skew-symmetric (n,n)-clans. We describe bijections between such clans, certain weighted lattice paths, and pattern-avoiding signed involutions, and we give a cell decomposition of the symmetric space in terms of collections of clans called sects. The largest sect with a conjectural closure order is isomorphic (as a poset) to the Bruhat order on partial involutions.
   
5. The genesis of involutions (polarizations and lattice paths (with Mahir Bilan Can – 2019)
   Discrete Mathematics, Volume 342, Issue 1, January 2019, Pages 201-216. https://doi.org/10.1016/j.disc.2018.09.026
   Abstract: The number of Borel orbits in polarizations (the symmetric variety SL(n)}/S(GL(p)x GL(q))) is analyzed, various (bivariate) generating functions are found. Relations to lattice path combinatorics are explored.
   
6. Counting Borel Orbits in Symmetric Varieties of Types BI and CII (with Mahir Bilan Can – 2018)
   Arnold Mathematical Journal, DOI:10.1007/s40598-018-0092-3
   Abstract: This is a continuation of our combinatorial program on the enumeration of Borel orbits in symmetric varieties of classical types. Here, we determine the generating series the numbers of Borel orbits in SO(2n+1)/S(O(2p)xO(2q+1)) (type BI) and in Sp(n)/Sp(p)xSp(q) (type CII). In addition, we explore relations to lattice path enumeration.