The 2006 Mathematics REU Program Research Activities Eight students participated in the 2006 mathematics REU program at the University of Akron. One group of four independently investigated problems related to wreath product finite p-groups. A second group of four students solved a real-world cryptography problem based on a collection of documents which form part of a correspondence between the sixteenth-century kings of Spain and their ambassadors in Italy (Details below.). The directors were pleased with the students' research efforts. Wreath Products: Wreath Product p-Groups Directed by Dr. Jeffrey Riedl Student participants: Sarah Bockting (University of Evansville), Arran Hamm (Wake Forest University), Alex Landraitis (Boston College), Sam Ruth (Northwestern University) Prior to the REU program, Dr. Riedl established, for an arbitrary prime p, a natural bijective correspondence between the normal subgroups of a certain finite wreath product p-group and the subspaces of the vector space of p-by-p matrices over the field of p elements that are simultaneously invariant under two particular nilpotent linear transformations. These so-called "doubly-invariant'' subspaces are in fact the submodules of the regular module of the group algebra FG, where F is the field of p elements, and G is the direct product of two copies of the cyclic group of order p. Even for small primes p, enumerating and constructing all the doubly-invariant subspaces is an extensive and difficult computational problem in linear algebra. The REU students investigated the case of the prime p=5, and employed an elaborate procedure, developed by Dr. Riedl, to enumerate and construct all the doubly-invariant subspaces. The procedure began by breaking the problem into sub-cases, based on the number and arrangement of the "standard basis matrices" contained in each doubly-invariant subspace. This splits the general problem into 2p choose p cases, and for the prime p=5 this yields 252 cases. Some of the cases are handled quickly and easily, while others were long and complicated. In working on this problem, the students confronted systems of linear equations whose coefficients are actually variables. The students produced a virtually complete solution to the problem for the case p=5. The total was just under 600,000 subspaces, and the written form of their solution covers about 500 pages. Tilings: Self-Similar Tilings Directed by Drs. Adler and Palagallo Student participants:Rebecca Black (Swarthmore), William Huddleson (Notre Dame), Lisa Lackney (University of Akron), James Rohal (College of Woster) The REU 2006 team generalized a classical theorem of Bandt (2001), concerning the construction self-similar periodic tilings, from Euclidean space to nilpotent Lie groups having certain rationality properties. The generalization to nilpotent Lie groups was natural extension since they are a class of groups on which we can find a metric; an automorphic dilation structure; discrete cocompact subgroups (i.e., lattices); and a measure. The notion of a periodic tiling makes sense in Euclidean space because of the existence of lattices. Moreover, we know how to transform any lattice into any other via a linear transformation. In order to construct tilings in nilpotent groups, we need to have some understanding of the kind of lattices they contain, and which lattices can be transformed into others via automorphisms. The REU team classified all such automorphisms of Heisenberg groups. They applied this to classify all lattices in the case where the group is three dimensional. One reason for interest in self-similar, periodic tilings is their connection with wavelets.