Research

Current

Currently, I am working on problems related to the combinatorial and topological structures of simplicial complexes. Here are my recent papers and preprints:

• Planar ternary graphs, flag spheres, and Delannoy polynomials. With Margaret Bayer, Richard Danner, Thiago Holleben, and Marie Kramer. Submitted. arXiv:2509.21705

Abstract: In 2022 Kim showed when a graph G is ternary (without induced cycles of length divisible by three), its independence complex Ind(G) is either contractible or homotopy equivalent to a sphere. In this paper, we show that when Ind(G) is homotopy equivalent to a sphere of dimension dimInd(G), the complex is Gorenstein. Equivalently, G is a 1-well-covered graph. This answers a question by Faridi and Holleben.

We then focus on the independence complexes of Gorenstein planar ternary graphs. We prove that they are boundaries of vertex decomposable simplicial polytopes. We show that the transformations among these flag spheres using edge subdivisions and contractions can be modeled by the Hasse diagram of the partition refinement poset. In addition, their h-polynomials are products of Delannoy polynomials and thus real-rooted. Finally, we demonstrate a way to construct nonplanar Gorenstein (1-well-covered) ternary graphs from planar ones.

• Reconstructing a shellable sphere from its facet-ridge graph. To appear in the Israel Journal of Mathematics. arXiv:2401.04220

Abstract: We show that the facet-ridge graph of a shellable simplicial sphere Δ uniquely determines the entire combinatorial structure of Δ. This generalizes the celebrated result due to Blind and Mani (1987), and Kalai (1988) on reconstructing simple polytopes from their graphs. Our proof utilizes the notions of good acyclic orientations from Kalai's proof as well as k-systems introduced by Joswig, Kaibel, and Körner.

• The Nevo--Santos--Wilson spheres are shellable. Electronic Journal of Combinatorics 31 (2024).

Abstract: Nevo, Santos, and Wilson constructed 2^Ω(N^d) combinatorially distinct simplicial (2d−1)-spheres with N vertices. We prove that all spheres produced by one of their methods are shellable. Combining this with prior results of Kalai, Lee, and Benedetti and Ziegler, we conclude that for all D>=3, there are 2^Θ(N^⌈D/2⌉) shellable simplicial D-spheres with N vertices.

This is an improvement from my previous result that these spheres are contructible. Here is a poster I made on that result for the Santander Workshop on Geometric and Algebraic Combinatorics:

Past

In summer 2020, I studied symplectic embeddings problems in the Virtual BeECH Group (consisting of Jo Nelson, Morgan Weiler, Leo Digiosia, Haoming Ning, and me). This led to the following paper:

Symplectic embeddings of four-dimensional polydisks into half integer ellipsoids. Journal of Fixed Point Theory and Applications 24, 69 (2022)

Here is a video recording of our talk on this result: