An example of the affine reflection group \(\tilde{A}_3\).

The graph of valid plane trees for sequence \(A\bar{A}A\bar{A}\bar{A}A\bar{A}A\).

The modified Young diagram for \(w_0 \in B_n\).
The only places that \(s_1s_2\cdots s_n\) appears as a subword of \(w_0\) are contained in the shaded boxes.

A variety of combinatorial objects

I like research projects that let me count fun mathematical objects: plane trees, Young diagrams, graphs, root systems, matrices, even strands of RNA. My work bridges several mathematical worlds including: algebra, algebraic geometry, combinatorics, and molecular biology.

Some of these problems have let me to use 3D printing to visualize the objects I study. Links to these projects along with photos and descriptions are on the coding page.

The pictures on the left are illustrations of some of the objects I care about. You can find more information about them in the papers below, or better yet, ask me about them!

Research with students

Amaechi Abuah ('21), me, Charles Yang ('19), and Gabrielle Kerbel (MHC '20)
Algebraic Splines Research Group, Summer 2018.

In the summer of 2018, I worked with three students on algebraic splines. What can we say about splines over a general ring with unity? Does the supersmoothness property for polynomial splines extend to algebraic splines? What sorts of algorithms will work for finding splines over general rings?

If you are a student interested in working with me, check out these problems to get you started thinking about the sorts of things I think about.

Papers and preprints

The Containment Poset of Type \(A\) Hessenberg Varieties.
E. Drellich, Submitted.

Geometric combinatorics and computational molecular biology: Branching polytopes for RNA sequences.
E. Drellich, A. Gainer-Dewar, H. Harrington, Q. He, C. Heitsch, and S. Poznanovic
Algebraic and Geometric Methods in Discrete Mathematics. Contemporary Mathematics 685 (2017), 137-154.

Valid plane trees: Combinatorial models for RNA secondary structures with Watson-Crick base pairs.
F. Black, E. Drellich, and J. Tymoczko
Accepted to SIAM J. of Discrete Mathematics.

A module isomorphism between \(H^*_T(G/P)\otimes H^*_T(P/B)\) and \(H^*_T(G/B)\).
E. Drellich and J. Tymoczko
Communications in Algebra, 45, (2017), no.1, 17-28.

Monk's Rule and Giambelli's Formula for Peterson Varieties of All Lie Types.
E. Drellich
Journal of Algebraic Combinatorics, 41 (2015), no. 2, 539-575.

Combinatorics of Equivariant Cohomology: Flags and Regular Nilpotent Hessenberg Varieties.
Doctoral Dissertation.