Combinatorics and Discrete Mathematics
Discrete mathematics is perhaps the fastest growing area of modern mathematics. It has a wealth of real world applications, especially in computer science, which have greatly contributed to its rapid growth. The Mathematics Department has an active research group in combinatorics, the central field of discrete mathematics. The faculty in our Combinatorics Group work in a variety of areas including algebraic combinatorics, discrete and combinatorial geometry, and graph theory.
Mark Ramras works in graph theory. He studies various independence and domination parameters, edge and vertex decompositions, Hamiltonian cycles, and permutation routings for bipartite graphs, and more specifically, hypercube networks, which are of interest in computer science, as they are the architecture used for parallel processing supercomputers.
Discrete and combinatorial geometry is mainly represented by Egon Schulte. He studies discrete geometric structures such as polytopes, polyhedra, complexes, and tilings, as well as their geometric, combinatorial, and algebraic symmetries. For example, a long-standing collaboration with Peter McMullen on the modern theory of regular polytopes has a few years ago resulted in a comprehensive research monograph, entitled "Abstract Regular Polytopes". More recent work studies the fascinating phenomenon of chirality in polytope theory, and explores polytopes and reflection groups over finite fields.
Andrei Zelevinsky explores topics in algebraic and polyhedral combinatorics most closely related to representation theory and algebraic geometry. For instance, his study of multivariate discriminants and resultants in computational algebraic geometry (joint with I. Gelfand and M. Kapranov) led him to the discovery of an important new class of convex polytopes called secondary polytopes. More recently he has discovered and explored new combinatorial structures arising in the theory of cluster algebras.
The research of several other Department members includes work on topics closely related to combinatorics: Maxim Braverman (polytopes and toric varieties), Anthony Iarrobino (combinatorial aspects of Hilbert schemes), Venkatraman Lakshmibai (Coxeter groups and the geometry of Schubert varieties), Mikhail Malioutov (combinatorial search problems), David Massey (hyperplane arrangements and singularities), Alexandru Suciu (combinatorics and topology of hyperplane arrangements), Jonathan Weitsman (problems involving analysis and combinatorics of convex polytopes), and Jerzy Weyman (combinatorial aspects of algebraic groups and quivers).