Combinatorial problems arising in finite group theory, 3-manifold Topology and other areas.
Shareshian works on combinatorial problems with applications in other fields. His joint work with R. Roberts and M. Stein involves using the theory of group actions on tree-like objects to investigate foliations of 3-manifolds. His joint work with R. Guralnick on actions of symmetric and alternating groups on k-sets is used to investigate monodromy groups of branched coverings of the Riemann sphere. His joint work with M. Wachs and others on graph and hypergraph complexes has applications in commutative algebra, finite group theory and knot theory. His work on topology of order complexes of intervals in subgroup lattices is intended to provide a new approach to a longstanding problem in universal algebra.
The principal investigator works on various problems in combinatorics. Roughly, combinatorics is the study of discrete (often finite) mathematical objects which admit fairly elementary descriptions but whose structure can be quite complicated. It is often the case that problems in other disciplines (such as computer science, electrical engineering and biology) and other more technical and abstract areas of mathematics can be reduced to combinatorial problems.
