Currently, I'm working under the direction of Professor Gretchen Ostheimer to investigate algorithms within the scope of infinite group theory. We (myself, Prof. Ostheimer and fellow students Camila Larsson & Daniel Dimijian) are developing a practical algorithm which accepts a set of matrices within GL(n,Q) that generate a polycyclic-by-finite group, and determines if Q^n is irreducible as a Q[G]-module. The computer algebra software GAP has been tremendously useful in computing various examples so far, which has helped us visualize the general function of the algorithm.
As an aside, I'm also intrigued by a difficult problem which arose in a Abstract Algebra 2 class discussion. Namely, given a set of maximal chain lengths, find the smallest partial ordering which admits this exact set. Here, the poset size is denoted by the number of elements in the partial ordering. We proved an upper bound of max(S)+|S|-1 and a lower bound of max(S)+log_2(|S|) for a given set S. I'm also looking to generalize possible formulations for a multi-set input, which allows for chain length repeats. The multi-set case is particularly illuminating because of its connections to lattice theory, set combinatorics, and number theory. As a trivial example, we can apply Sperner's theorem and the Lubell-Yamamato-Meshalkin inequality as lower bounds by utilizing the trivial lemma that no maximal chain is contained within another.
This summer, I'll be attending the REU program hosted at the Rutgers DIMACS facility, where I'll be studying graph theory. These are tentative plans.