Commutative algebra and its interaction with combinatorics is a popular topic among researchers. One such interaction occurs between polynomial rings and graph theory. An example of this is the binomial edge ideal: given a finite simple graph G, we build an ideal B(G) by declaring its generators to be the collection of all binomials x_iy_j-x_jy_i such that {i,j} is an edge of G. An interesting question is, what algebraic properties can be read from B(G) in terms of the combinatorial properties of G, or vice-versa? A nice introduction to this topic can be found here.
Recent developments on the algebraic side of things involve studying sequences of ideals that are related by some symmetries, and their "long term behavior". The goal of this project is to consider sequences of related binomial edge ideals and attempt to transfer ideas over to the graph theory side of things. For example, what does a sequence of "related" graphs look like in this context? Can we find properties of these graphs that start to hold (or break) after you go out far enough in the sequence? The nature of this project lends itself nicely to examples, computation and coding.
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