While there is so much in math that is interesting and engaging, time is merely an illusion that I have yet to see past. Since I cannot find enough time to work in every area, I will have to focus on just a few.
My research interests are primarily in Potential Theory, Discrepancy Theory, and Approximation Theory, although I am also interested in Additive Combinatorics, Convex Geometry, and in general Harmonic Analysis.
I have also worked on several projects in Graph Theory in my undergraduate and early graduate career, and have some interest in the subject.
Some of my current projects include:
- Developing potential theory for kernels with more than two inputs.
- Developing a better understanding of repulsive-attractive potential, and minimizers of their energies, with a particular focus on p-frame energies.
- Determining analogues of the Stolarsky Invariance Principle and their applications.
Publications (Submitted, Accepted, Published):
- Energy on Spheres and Discreteness of Minimizing Measures (arXiv), Submitted, 2019 (with D. Bilyk, A. Glazyrin, J. Park, and O. Vlasiuk)
- Optimal Measures for p-frame Energies on Spheres (arXiv), Submitted, 2019 (with D. Bilyk, A. Glazyrin, J. Park, and O. Vlasiuk)
- On Subgraphs with Prescribed Eccentricities, Submitted, 2018 (with K. Bragan, P. Dankelmann, M. DeVilbiss, D. Erwin)
- On the Fejes Tóth Problem about the Sum of Angles Between Lines (arXiv), Proceedings of the AMS, 147(1), 2019, 51-59 (with D. Bilyk)
- The Maximum Size of (k,l)-Sum-Free Sets in Finite Cyclic Groups (arXiv), Bulletin of the Australian Mathematical Society, 99(2), 2019, 184-194 (with B. Bajnok)
- The Edge Grundy Number of the Regular Turan Graphs, Bulletin of the Institute of Combinatorics and its Applications, 84, 2018, 45-52 (with M. DeVilbiss and P. Johnson)
- Rainbow Connectivitiy in some Cayley Graphs, The Australasian Journal of Combinatorics, 71(3), 2018, 381-393 (with S. Bau, P. Johnson, E. Jones, and K. Kumwenda)
- Stolarsky Principle and Energy Optimization on the Sphere (arXiv), Constructive Approximation, 48(1), 2018, 31-60 (with D. Bilyk and F. Dai)
- Connected Minimum Secure Dominating Sets in Grids, AKCE International Journal of Graphs and Combinatorics, 14(3), 2017, 216-223 (with J. Barnett, A. Blumenthal, P. Johnson, C. Jones, and E. Mujuni)
- The Edge Grundy Number of some Graphs, International Journal of Mathematics and Computer Science, 12(1), 2017, 13-26 (with L. Anderson, M. DeVilbiss, S. Holliday, P. Johnson, A. Kite, and J. McDonald)
- On the Minimum Size of Signed Sumsets in Elementary Abelian Groups (arXiv), The Journal of Number Theory, 159, 2016, 384-401 (with B. Bajnok)
- The Minimum Size of Signed Sumsets (arXiv), The Electronic Journal of Combinatorics, 22(2), 2015, (P2.50) 1-17 (with B. Bajnok)
- Note on the Secure-Domination Number of a Graph, Bulletin of the Institute of Combinatorics and its Applications, 74, 2015, 113-119 (with M. DeVilbiss, B. Fain, and P. Johnson)