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.