My research interests lie primarily in the connections between different areas of mathematics and between mathematics and the outside world. I am a combinatorialist, fascinated by the discrete, with graph theory being a first love. Combinatorial PolynomialsPolynomials arise from graphs in a variety of different ways, including:
My particular interest lies in the nature and location of the roots of such polynomials. The distribution of the roots in the complex plane can inform you of combinatorial propertiues of the underlying coefficients, including unimodality. Often there are underlying simplicial complexes for which the polynomials are simple evaluations of fpolynomials (or face polynomials) of the complex, and this puts many of these polynomials in a common framework. The images generated by the roots of various graph polynomials can be quite beautiful, and sometimes fractal in nature. Also, when the underlying simplicial complexes have a particular property known as shellable, then there is a deep connection to commutative algebra. The study of the associated ideals are not only interesting in their own right, but also point to interesting questions involving Grober bases over finite fields. Other polynomials of interest are
Well Covered Graphs and Vector Spaces A graph is well covered if every maximal independent set has the same size. There is much known about the structure of such graphs. In conjuction with Richard Hoshino I looked at well covered circulants (those are graphs on the integers mod n where adjacency corresponds to the difference being in some set S, not containing 0, with S = S). One can extend the notion of well coveredness algebfraically as follows. Suppose G is a graph and k a field. A weighting is a function from the vertex set of G into k; such a weighting is a well covered weighting if the sum of the weights over any maximal independent set is constant. The weightings form a vector space over k, and questions of dimension and bases are intriguing. Mathematics and Music There are a number of ways that mathematics intersects with musicand I have applied mathematics to a number of areas of music:
