Ryan Martin

The focus of my research is extremal graph theory and random combinatorial structures. I have also worked in a variety of other areas, including intersecting hypergraphs, the theory of positional games and Ramsey theory. Combinatorics, and particularly graph theory, has a wide variety of applications. My own research in the edit distance of graphs has applications in biology and computer science, where networks evolve and change via adding or destroying connections between nodes. Vertex-identifying codes in graphs are an efficient way of monitoring a large network. Graph theory in general has deep connections to other areas of pure mathematics and their applications, such as additive number theory, harmonic analysis, ergodic theory and topological dynamics, as well as computational geometry with applications in dual source codes, and the time complexity of parallel computation. I use a number of tools in my research, notably probabilistic methods and, most prominently, Szemerédi’s regularity lemma. I have used these and other techniques to address questions related to graphs, hypergraphs and combinatorial structures. The following is a summary of some major facets of my research. I cannot cover all of the papers adequately, but I hope to summarize some of the major themes. Reference numbers correspond to the numbering of the papers in my CV and the reference numbers are hyperlinks to a preprint of the referenced manuscript.