Arrangements of Hyperplanes and Combinatorial Constructions in Topology

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My research involves interactions between combinatorics and techniques from areas of algebra, topology, and geometry.******Combinatorics is a branch of mathematics which sits at the foundation of computer science and operations research. It involves the study of discrete structures as well as of counting problems. Many interesting objects in other areas of mathematics are fundamentally combinatorial in nature, so methods of combinatorics appear naturally in understanding families of "test" objects such as toric varieties and hyperplane arrangements. These can be used as accessible examples to help identify and understand general mathematical phenomena. ******In a different direction, the well-developed techniques of algebra and geometry can sometimes answer combinatorial questions and offer deep reasons for observed or conjectured behaviour of objects in discrete mathematics. For example, if the answer to a counting problem is a number which in itself shows no extra structure, one might find that this number is, in fact, some evaluation of a polynomial. Perhaps the polynomial has coefficients which themselves may be interpreted as volumes of certain polyhedra, or they count the number of high-dimensional "holes" in a geometric object which could be built from the original problem. In this way, one may obtain a better understanding of a combinatorial object, by viewing it as a shadow of something with much more structure and, one hopes, properties which are better known or at least more easily analyzed.