Daniel K. Nakano

Representation theory has emerged as a central area of modern mathematics with connections to combinatorics, algebraic geometry, topology, number theory, along with applications to chemistry and physics. Representations are mappings, regarded as snapshots, of complicated algebraic objects (such as groups and Lie algebras) to an array of numbers (matrices). These realizations by matrices encode local data that can yield deep insights into these complicated algebraic objects.

Recent developments have included using the representation theory of Lie groups to better understand signal processing and the translation of the P versus NP Problem in computer science via geometric complexity theory into questions about decomposing representations for the general linear group.

My research is at the interface between representation theory, geometry and homological algebra. Collecting all the representations for a certain algebraic object forms a tensor triangulated category. Methods from homological algebra can be used to build a bridge between tensor triangulated categories and geometric objects. Uncovering this “hidden geometry” often leads to new insights about the algebraic object and its representations.