Research
The roots of my research began in graph theory (combinatorics) and topology. Specifically, I studied the homology of a simplicial complex constructed from a graph. This work was published in the paper Nerve Complexes and Circular Arcs with Henry Adams, Michal Adamaszek, Florian Frick, and Chris Peterson. See the journal Discrete & Computational Geometry 56 (2016), 251-273 or click here for access.
DNA Self-Assembly
More recently, I have been studying DNA self-assembly design strategies using concepts from graph theory. This work began as a project at REUF 2017 (see here) in collaboration with Amanda Harsy (Lewis University), Leyda Almodovar Velazquez (Stonehill College), Jessica Sorrells (Converse University), and Jo Ellis-Monaghan (University of Amsterdam).
DNA self-assembly, and self-assembly in general, is a rapidly advancing field. Examples of synthetic DNA molecules that have been designed to self assemble into nanostructures include nanoscale arrays, polyhedra, arbitrary graphs, a variety of DNA and RNA knots, and the first macroscopic self-assembled 3D DNA crystals. This has led to molecular scaffoldings made of DNA which have wide-ranging potential, such as containers for the transport and release of nano-cargos, templates for the controlled growth of nano-objects, biomolecular computing, biosensors, and drug-delivery methods.
Our group has submitted a paper Computational complexity and pragmatic solutions for flexible tile based DNA self-assembly which is currently under review. A copy of the paper may be found on arXiv: https://arxiv.org/abs/2108.00035
Additionally, we have a repository of results for certain classes of graphs which can be found on CSUSB's Scholarworks website: https://scholarworks.lib.csusb.edu/mathematics-publications/1. The repository includes results for Platonic solids, square lattice graphs, and triangle lattice graphs.
Student Research
Undergraduate Research
I have worked with about 10 undergraduate students on independent studies and research projects. Projects have included the areas of Cryptography and Graph Theory (graph coloring, Hamilton cycles, and the hat problem). Most recently, I have been working with students on open problems in DNA self-assembly that are accessible at the undergraduate level. A background in introductory graph theory and in linear algebra is helpful but not necessary.
A copy of Gabriel Lopez's paper on Self-Assembling DNA Complexes with a Wheel Graph Structure can be found on arXiv here: https://arxiv.org/abs/2302.13014
Graduate Research
I have worked with three graduate students on Master's theses in the area of DNA Self-Assembly (see below). One thesis focused on the interplay of linear algebra and related design strategy questions. The other theses model DNA complexes using specific graph structures. Copies of theses can be found through the CSUSB ScholarWorks website.
Hytham Abdelkarim - DNA Self-Assembly of Trapezohedrial Graphs (graduated summer 2023)
Andrew Lavengood-Ryan - DNA Complexes of One Bond-Edge Type (graduated spring 2020)
Ernesto Gonzalez - Tile Based Self-Assembly of the Rook's Graph (graduated spring 2020)
Emelin Sibrian-Marquez - DNA Self-Assembly of the King's Graph (in progress)
If you are interested in working on a research project, please contact me.
Summer@ICERM - REU
In summer 2023, I co-organized and co-facilitated an REU at ICERM (Institute for Computational and Experimental Mathematics) in Providence, Rhode Island. The program focused on Mathematical Modeling of DNA Self-Assembly. My co-leaders and I mentored 18 undergraduate students from universities across the United States over the span of 8 weeks. Students split into five groups which each focused on different computational problems in the area of DNA self-assembly. Each group has a paper in progress (as of fall 2023). For more details, please see the program page: Summer@ICERM