Research
Much of the work of my dissertation concerns unavoidable immersions of finite and infinite graphs. The finite results are contained in a paper about 30 pages long, which has been submitted and can be viewed on the arXiv. I was awarded the inaugural Bogdan Oporowski Prize for the contributions made in this paper. The infinite results are currently being written up in another paper, which I hope to submit by December.
To see a version of my Research Statement with more detail than I submitted with my applications, you can expand the document to the right.
research_statement_detailed.pdfResearch Interests
I'm primarily interested in structural and extremal graph theory, particularly in Ramsey-type "unavoidable substructure" results. I'm also interested in similar results on structures other than graphs, especially matroids and hypergraphs.
Outside of Ramsey theory, I like recognition problems: "Given a graph G we can define an auxiliary graph f(G). Given an arbitrary graph H, how can we decide if H =f(G') for some graph G'?"
Another interest of mine lies in analogizing certain graph-theoretic concepts, like cographs, to the realm of hypergraphs. I have fun investigating what generalizes nicely and what changes drastically when moving between graphs and hypergraphs.
My last primary interest is in design theory. This was the topic of my Master's thesis at McNeese State University. I find the arguments here are often elegant, and this field is especially well-suited for the introduction of younger students to research (as was the case for me!). There are lots of open problems throughout the field of combinatorial design which I'm interested in tackling someday.