Below is a list of my research interests, student projects, as well as recent publications.  If you are an undergraduate interested in mathematical research, please contact me for more information!

Research Interests

• Analysis, Differential Equations, Difference Equations

  • Using functional analysis techniques (fixed point theorems, variational techniques) to establish existence and uniqueness results for various classes of nonlinear boundary value problems.

• Mathematical Models of Cancer

  • Exploring local and global stability of steady states for various ODE cancer growth and treatment models.

  • Developing reaction-diffusion (PDE) models for polyp growth in colorectal cancer.

  • Incorporating the cancer stem cell hypothesis into existing cancer models to observe novel dynamics.

Other General Interests

• Neural Networks, Mathematical Models of Humor, Relativity, Cosmology, Chaos Theory, Math in Music

Undergraduate Research Projects

I have had the pleasure of working with students on a variety of projects that focus on creating mathematical models of real-world phenomena ranging from cancer growth to the humor response:

• National Research Experience for Undergraduates Program (NREUP) - Summer 2013 - Mathematical Models of Cancer Networks - with Dr. Joe Rusinko, 3 Winthrop undergraduates, and 1 Western Carolina undergraduate

• McNair Scholars Program - Summer 2013 - A Reaction-Diffusion Model of Colorectal Cancer - with 1 Winthrop undergraduate

• Senior Honors Thesis - Fall 2012/Spring 2013 - An Artificial Neural Network Modeling the Humor Response - with 1 Winthrop undergraduate

• Summer Research Experience in Mathematics Program - Summer 2012 - Mathematical Models of Colon Cancer - with Dr. Joe Rusinko, Dr. Kristen Abernathy, and 10 Winthrop undergraduates

Recent Publications

• G. Epelle and Z. Abernathy (2013), "A Mathematical Model of Treatment of a Linear Cancer Network with Immunotherapy", Submitted, International Journal of Undergraduate Research and Creative Activities.

• J. Rodriguez and Z. Abernathy, "On the solvability of nonlinear Sturm-Liouville problems", Journal of Mathematical Analysis and Applications, 387 (2012), 310-319.

• J. Rodriguez and Z. Abernathy, "Nonlinear discrete Sturm-Liouville problems with global boundary conditions", Journal of Difference Equations and Applications, 18 (2012), 431-445.

• Z. Abernathy and J. Rodriguez (2011), "Existence of Periodic Solutions to Nonlinear Difference Equations at Full Resonance", Submitted, Journal of Difference Equations and Applications.