The goal of most graduate special topics classes is to give the students an in-depth knowledge of important, technical subject matter not covered in the graduate core. The principle behind this special topics course is quite different: the focus is on exceptional structures—certain distinguished examples found (in different guises) in virtually every branch of mathematics, and especially in algebra, discrete mathematics, and geometry. Often the key to uncovering deep and unexpected connections between different disciplines is first to understand the equivalence between examples from each.
The course will focus on three main "sources" of exceptional structures: division algebras, the ADE classification, and the Leech lattice. There are many more structures we would like to discuss than time to cover them within a quarter class. Nevertheless, we expect to select topics from among: the quaternions, the octonions, exceptional Lie groups/algebras, root lattices, Jordan algebras, non-Desarguesian projective planes, Golay codes, the Leech lattice, Steiner systems, finite simple groups, the Mathieu and Conway groups, sphere packings and spherical codes, horosphere packings, Kodaira's theory of elliptic surfaces, Mordell-Weil lattices, ADE singularities, DuVal singularities, Arnold's Strange Duality, quivers and the McKay correspondence, reflection groups of Euclidean and Lorentzian lattices, exceptional holonomy manifolds, etc.
Depending on the number and background of the students attending, most likely the first 2/3 of the course will consist of lectures by the instructors and the last 1/3 of supervised student presentations on topics of particular interest to each student.
Prerequisites: The course should be accessible to students who have completed the first-year graduate Algebra. Some basic knowledge of algebraic geometry would be also be useful; but an interested student with no background in algebraic geometry can certainly see us about outside reading to fill in any gaps.