Date: Tuesday October 8, 2019
Location: Rocky 307
Title: Combinatorics and Group Symmetry in [Hilbert Space] Frames
Frames are collections of vectors in Hilbert spaces which have reconstruction properties akin to orthonormal bases. In order for such a representation system to be robust in applications, one often asks that the vectors be geometrically spread apart; that is, the pairwise angles between the lines they span should be as large as possible. It ends up that structures in combinatorics, like difference sets and balanced incomplete block designs (BIBDs), can be used in different ways to construct optimal configurations. Furthermore, the linear dependencies of the vectors are often encoded as BIBDs. The orbit of a vector under a group action sometimes also yields an optimal configuration. There are infinite classes of frames, including the so-called Gabor-Steiner ETFs, which have both group symmetry and a combinatorial construction. In this talk, these and other connections between frames and algebra, geometry, and combinatorics will be presented. A couple of open problems in frame theory and quantum information theory will also be discussed, including Zauner's conjecture. Frames, difference sets, and BIBDs will be defined and explained, making the talk accessible to a more general audience.