Alex Nowak, Iowa State "Representation theory for certain combinatorial designs"
Abstract: In combinatorial design theory, Mendelsohn triple systems (sometimes referred to as cyclic triple systems) generalize Steiner triple systems, and in universal algebra, quasigroups are the so-called nonassociative groups. Mendelsohn triple systems may be realized as idempotent, semisymmetric quasigroups, which are quasigroups satisfying the identities x2 = x and y(xy) = x. Exploiting this universal algebraic description, I will present a theory of modules over Mendelsohn triple systems. As in the group case, the module theory for these nonassociative algebras prompts questions regarding extension. I will address some of these extension problems, and if time permits, I will also discuss the interplay between representations of Steiner and Mendelsohn triple systems