Topology and algebraic geometry. Computational aspects of homotopy theory and its connections with geometry, number theory, and quantum field theory. Specifically, uncovering higher genus generalizations of elliptic cohomology suggested by the exceptional geometry of the Cayley plane (and categorifying moduli spaces of theta characteristics). The ongoing computation of the homotopy type of the string bordism spectrum MO<8> at the prime 3, based on computer-assisted computations of its BP-homology, considered as a Hopf ring. Novel uses of differential geometry in data analysis. Computational and geometric aspects of bookbinding and music composition.