For a complex hyperplane arrangement, the cohomology ring of the complement depends only on the combinatorics of the arrangement, and this cohomology ring has an explicit expression as the quotient of an exterior algebra E. One may study such rings by studying their resonance varieties, collections of points corresponding to (nontrivial) zero divisors. Viewing these cohomology rings as modules over E, the Chen Ranks Theorem expresses some of the graded Betti numbers of the ring in terms of the first resonance variety. Inspired by this result, I will define a collection of varieties consisting of the points in the resonance varieties that "contribute to the Betti numbers," and state a theorem I proved computing these for all square-free E-modules. I will conclude with an application of the result to cohomology rings of hyperplane arrangements.

In this talk I will give a quick overview on both diffeology, and sheaves of modules on smooth manifolds. Both diffeological vector bundles and sheaves of modules, can be thought of as generalizations of the notion of a vector bundle on a manifold. At first glance, the two approaches seem quite different, but I will show that one can construct a natural functor from the category of sheaves of modules to the category of diffeological vector bundles. We will then see that, under some fairly mild assumptions, this functor is actually an equivalence of categories. In the last part of the talk I will give some examples where understanding this correspondence has proven to be useful.

We introduce Stanley's chromatic symmetric function (CSF) for a simple graph, which is a generalization of the chromatic polynomial. Stanley gave the expansion of CSF in various bases of space of symmetric functions, and asked whether a tree is determined by its respective CSF. We describe a four term relation satisfied by CSF and give a recursive formula to expand it in star bases.