Research Interests

My research focuses on algebraic coding theory and cryptology, using algebraic and combinatorial methods to study error-correcting codes and cryptographic systems. I also involve undergraduates in projects that connect abstract algebra, combinatorics, and coding implementations. My earlier work explored representation theory, Schubert varieties, and Bruhat order posets, which continue to shape my mathematical perspective.

Publications and Preprints

1. Lexicographic shellability of sects with Aram Bingham: We show that the sects of the type-symmetric variety form shellable posets. We continue to believe that Bruhat posets on homogeneous spaces such as symmetric varieties should, in general, be shellable. As a partial result, we found it a pleasant surprise that a bijection between sect orbits and rook placements in partition shapes, together with ideas from the work of our advisor Mahir Can, allowed us to establish shellability in this setting. Interestingly, this also appears to be the first time that the shellability of the poset of matrix Schubert varieties has been explicitly demonstrated (The Electronic Journal of Combinatorics, 2025).
2. Nearly Toric Varieties of Type \(\mathsf{A}\) with Mahir Can: We combinatorially characterize a novel family of Schubert varieties defined by the conditions of minimum torus codimension one and minimum Borel codimension zero. We term this collection nearly toric varieties because they constitute a family of spherical varieties that closely resemble toric varieties. Our work proceeds in two stages: First, we lay the combinatorial groundwork by explicitly identifying the family of nearly toric Schubert varieties. Second, we establish both quantitative (enumerative) and qualitative (generators and relations) results concerning the singularities of these varieties (Turkish Journal of Mathematics, 2025).

Invited Talks

1. Shellability of symmetric spaces and Bruhat orders. SIAM TX-LA, University of Louisiana at Lafayette, LA, USA, Nov 3-5, 2023.
2. Enumerating spherical Dyck paths and smooth nearly toric varieties. DiscreteMath, Universidad Nacional, Bogotá D.C., Colombia, Oct 9, 2023.
3. Spherical partition Schubert varieties and Dyck paths. SRAC, NOLA, USA, March 24-26, 2023.
4. Dyck paths and nearly toric Schubert varieties. JMM, Boston, USA, January 4-7, 2023.
5. Ding and Schubert varieties. SMM, Universidad De Guadalajara, Jalisco, México (virtual-Mini-Plática).
6. Contar nos hace la vida más fácil. COCITEI, Oaxaca, México (virtual-slides).
7. \(Q_p\) Spaces on Hyperbolic Riemann Surfaces. Encuentro de Sociedades de Matemáticas Colombiana y Mexicana, Universidad del Norte, Colombia (slides).

Contributed Talks

1. Nearly toric Schubert varieties and Dyck paths. \(13^{th}\) Southeastern Lie Theory Workshop, Raleigh, NC, USA, May 12-14, 2023 (slides).
2. Dyck paths and nearly toric Schubert varieties. CombinaTexas, College Station, Texas, USA, April 22-23, 2023.
3. Ding and Schubert Varieties. Graduate Student Colloquium at Tulane university, USA (slides).
4. A naive Introduction to Affine Schemes. Graduate Student Colloquium at Tulane university, USA (slides).
5. From Differential Geometry to Lie Theory. Graduate Student Colloquium at Tulane university, USA (virtual-slides).
6. Manifolds and their applications. Graduate Seminar at IPN, México (slides).

Posters

1. Lexicographic shellability of sects. CAAC , LACIM (UQAM), Montreal, Canada, Jan 26–28, 2024 (poster).
2. Spherical Dyck paths. Permutation Pattern, University of Burgundy, Dijon, France , July 3–7, 2023 (poster).
3. Spherical partition Schubert varieties and Dyck paths. SLAM, University of North Texas, March 4–5, 2023.
4. Spherical partition Schubert varieties. PRIMA congress, Vancouver, Canada, December 4-9, 2022.