ALGEBRA
Broadly, my interests are in algebraic deformation theory and homological algebra. More recently, I have been concerned with the construction of new families of Frobenius seaweed algebras. These algebras arise naturally as subalgebras of semisimple Lie algebras and are deeply connected to algebraic deformation and quantum group theory through their relationship to the classical YangBaxter equation.
GEOMETRY
I am interested in properties of hypersurfaces of revolution in various ambient spaces. In Euclidean spaces, I have been led to the construction of objects which are called equizonal ovaloids. These objects satisfy a generalized equal area zones property which characterizes the sphere in ambient dimension three. The equizonal ovaloids are dual to important minimal hypersurfaces of revoution. We have classifed these equizpnal objects in all the space forms, and used them to develop another class of geometric objects called Archimedian hypersurfaces. In the coming year, I will be looking at the algebraic case where the ambient space is complex or quaternionic projective space.
GRAPH THEORY
In combinatorics, Ramsey's theorem states that in any coloring of the edges of a sufficiently large complete graph (that is, a simple graph in which an edge connects every pair of vertices), one will find monochromatic complete subgraphs. For 2 colours, Ramsey's theorem states that for any pair of positive integers (r,s), there exists a least positive integer R(r,s) such that for any complete graph on R(r,s) vertices, whose edges are colored red or blue, there exists either a complete subgraph on r vertices which is entirely blue, or a complete subgraph on s vertices which is entirely red. Here R(r,s) signifies an integer that depends on both r and s. It is understood to represent the smallest integer for which the theorem holds. Joint work with Magnant and Ryan has developed a generalization of Ramseys theory which concerns itslelf with what we have called preRamsey numbers.
