Daniel A. Ramras

• Clay Kellogg: Representations of crystallographic groups .
  

During Summer 2016, Clay studied irreducible representations of certain crystallographic groups, generalizing some examples from my article in Forum Math. (2014).

• Chris Neuffer: Crystallographic groups.

During Summer 2014, Chris studied crystallographic groups and the associated flat manifolds. The focus of the project was to find a characterization of orientable flat manifolds in terms of their fundamental groups. Chris showed that a flat manifold M is orientable if and only if each element in the fundamental group of M acts on the translation subgroup with positive determinant, and he spoke about his work at the Indiana REU conference (July 23, 2014). Chris is currently a Ph.D. student at IUPUI.

• Jonah Wyatt: Discrete Morse theory, Hom complexes, and homotopy groups of graphs.

Jonah studied applications of discrete Morse theory and homotopy theory of graphs to Hom complexes. For instance, discrete Morse theory can be used to study when two Hom complexes are homotopy equivalent. Jonah investigated discrete Morse functions associated to folds, following work of Kozlov. More recently, Jonah developed new combinatorial notions of homotopy groups for graphs, which should be closely related to homotopy groups of clique complexes and Hom complexes.

• Homology of Hom Complexes, by Mychael Sanchez. Rose-Hulman Undergrad. Math. J. 13 (2012), 212-223

Abstract: The hom complex Hom(G,K) is the order complex of the poset composed of the graph multihomomorphisms from G to K. We use homology to provide conditions under which the hom complex is not contractible and derive a lower bound on the ranks of its homology groups.

This project was completed during Mychael's junior and senior years at New Mexico State University, and was funded in part by my NSF grants, DMS-1057557 and DMS-0968766. Mychael is currently completing his Ph.D. in mathematics at the University of Illinois Urbana-Champaign.