Algebraic Methods in Combinatorics and Finite Geometry

University Of Delaware, Newark DE

Investigators

Abstract

The project's focus is on algebraic methods in combinatorics and finite geometry. Combinatorics is a fast-growing area of mathematics. It has a wealth of computational, scientific, and engineering applications, ranging from algorithm analysis to human genome sequencing to cellular phone technology. In combinatorics one considers discrete (as opposed to continuous) structures such as graphs, hypergraphs, designs, matroids, and finite geometries. One central task of this research project is to prove existential, enumerative, and constructive results concerning these structures. This research project is centered on algebraic invariants of various matrices arising in combinatorics and finite geometry. Typical matrices considered are incidence matrices of designs and finite geometries, higher inclusion matrices of set systems, and adjacency or Laplacian matrices of graphs. The invariants under investigation include modular ranks, spectrum, and Smith normal forms. The investigator intends to pursue several research directions. One direction is to apply these invariants to solve existential problems in design theory, extremal combinatorics, and finite geometry. Another direction is to use algebraic invariants for the purpose of distinguishing non-isomorphic combinatorial structures with the same parameters.