Algebra and combinatorics

My research is on algebraic structures called "rings". I use graphs and other such geometric structures to find algebraic properties of rings, and conversely I am interested in how the algebra of rings can affect the geometry of the objects I use. One technique that I apply often is to associate to a certain ring a graph, that is, a set of points some of which are connected by lines, or a hypergraph, which is a higher-dimensional counterpart of a graph. Using this technique I have been able to partially classify hypergraphs that correspond to rings with desirable properties from an algebraic point of view. My goal is to conduct deeper investigations in this direction and to find stronger classification techniques. The importance of this project is that many algebraic properties that are difficult to verify for a ring may be easier to verify once they are translated into properties of hypergraphs.