Talk Materials:

"Unit Circle" Animation
"Parabola" Animation
    These animiations show what the unit ball in R^2 and a parabola with vertex at the origin look like under different norms. The unit ball starts as a diamond in the 1-norm (also known as the Taxicab norm), rounds out to the familiar unit circle (achieved by the Euclidean norm) and approaches the unit square (achieved by the max-norm).

Pi Day Presentation - I gave this talk as a Pi Day presentation to a local MENSA chapter.

Research:

My research interests range from theoretical areas in functional analysis and differential equations to applied areas in cancer research and mathematical modeling. I am currently studying sufficiency conditions for unique solutions to Sturm-Liouville problems with nonlinear boundary conditions. I also am working with four undergraduates to mathematically model the effects of colorectal cancer. Below is a list of publications along with works in progress. If you're interested in my work, would like to look at a preprint, or are an undergraduate thinking about research in mathematics, please send me an email!

Functional Analysis/Differential Equations

  1. Sufficient conditions for multipoint boundary value problems (accepted by Differential Equations and Applications)
  2. Nonlocal boundary value problems for discrete systems (appeared Journal of Mathematical Analysis and Applications 2012)
  3. Existence of solutions to boundary value problems at full resonance (appeared Differential Equations and Applications 2011)
  4. On the solvability of nonlinear boundary value problems (appeared Differential Equations and Applications 2010)
  5. Existence and computation of multiple positive solutions of singular nonlinear boundary value problems (appeared Journal of Computational and Applied Mathematics 2010)

Cancer Research/Mathematical Modeling

  1. Introduction to mathematical modeling: cell growth in a colon crypt (work in progress with undergraduate students)

NetLogo Models

Netlogo website - Link will take you to the program's website where you can download NetLogo and explore other user generated models.

Colorectal cancer model
The purposes of this model are to compare a healthy colon crypt to a mutated colon crypt and demonstrate the effects of mutated stem cells on polyp growth. This model specifically focuses on bottom-up morphogenesis, which occurs when the stem cells have a mutation that causes the production of abnormal daughter cells. This model was created as part of an undergraduate research program during the summer of 2012 at Winthrop University. Authors of this program and accompaning lesson plans and worksheets include Winthrop faculty Kristen Abernathy, Zachary Abernathy, and Joseph Rusinko, and student researchers Katrina Harmon, Anna Johnson, Johnakin Martin, and Kirsten Stallings. Any questions should be directed to Kristen Abernathy at abernathyk@winthrop.edu.
        NetLogo file for colorectal cancer model
        Picture needed for program - Download the colon cancer program and picture to the same folder. Open NetLogo and choose to open the colon cancer program. As long as the colon cancer program and picture are in the same file, the program will work.
        Lesson plans and worksheets to accompany cancer model:
                1. Middle level mathematics course
                        Lesson plan
                        Worksheet
                2. Algebra I course
                        Lesson plan
                        Worksheet
                3. Introductory to Data Analysis course
                        Lesson plan
                        Worksheet