Research

Current

Currently, I am working on problems related to simplicial complexes. Specifically, I have been studying decomposability properties (such as shellability, contructibility, and partitionability) of simplicial spheres. Here are my recent preprints:

• Reconstructing a Shellable Sphere from its Facet-Ridge Graph. 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. arXiv:2305.03186

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 Dr. Jo Nelson, Dr. Morgan Weiler, Dr. Leo Digiosia, Haoming Ning, and me). This led to the following paper:

Symplectic embeddings of four-dimensional polydisks into half integer ellipsoids. JFTPA 24, 69 (2022)

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