Research Interests
NUMERICAL SEMIGROUPS
Drs. Ranthony Edmonds, Bethany Kubik, Christopher O'Neill, and I are beginning a project about numerical semigroups and atomic density - look for more to come on this in the future.
MAGIC LABELINGS
Dr. Bethany Kubik and I have done some exploring of magic labelings on certain graphs. A magic labeling of a graph is where each edge is labeled with a positive integer so that each vertex is associated with the sum of all incident edges, and this sum always yields the same value (i.e. the magic number). We seek to determine when certain graphs can be labeled with a magic labeling and what are the possible magic numbers associated with such a labeling.
GENERALIZED LAYERED CROWNS
I have worked on research with Dr. Rebecca Garcia, Dr. Pamela Harris, Dr. Bethany Kubik, and CPT Joseph Pederson regarding generalized layered crowns. For more information, I refer you to Rebecca Garcia's homepage for her research statement.
The work on generalized layered crowns expanded into dimension theory and a study of independent sets of a graph. As the research has morphed into a larger project, the collaboration has come to include Fidel Barrera-Cruz, Heather Smith, Libby Taylor, and Dr. William T. Trotter.
REPRESENTATION THEORY
My dissertation research involved deformation theory, which studies the behavior of mathematical objects, such as representations or modules, under small perturbations.
Quivers, which are directed graphs consisting of vertices and arrows, provide a combinatorial framework for the study of representations of algebras. You can see an example of a quiver pictured above. A set of relations of the quiver is a linear combination of paths (with coefficients in a certain field k) that begin and end at the same vertex. We can consider a quiver with relations and then combinatorially obtain diagrams like the one below, called components. The components are used to methodically find certain modules for which we determine universal deformation rings.
For more information on the above topic, I suggest the following reads:
Representation Theory of Artin Algebras by M. Auslander, I. Reiten, and S. Smalø
Blocks of Tame Representation Type and Related Algebras by K. Erdmann
On Auslander-Reiten components of blocks and self-injective biserial algebras by K. Erdmann and A. Skowronski