This special colloquium will host a faculty candidate for data science.
Speaker: Joey Iverson, Research Associate, Department of Mathematics, University of Maryland
Title: Optimal line packings from finite group actions
Abstract: Frames are spanning sets of vectors that allow for stable analysis and reconstruction of data in Hilbert space. In applications such as compressed sensing and quantum information theory, it is critically important to find examples of frames whose vectors are spread wide apart. In particular, we would like the interior angles between the lines they span to be as wide as possible, as measured by the coherence. This is an old problem, going back at least to the work of van Lint and Seidel in the 1960s, and it remains an active and challenging area of research today. In this talk we will present a new recipe for converting transitive actions of finite groups into tight frames, many of which have optimal coherence. The main idea is to use an association scheme as a kind of converter to pass from the discrete world of permutation groups into the continuous setting of frames. This process is easy to implement in a computer program like GAP. We will present several examples of optimally incoherent frames produced in this way, including the first infinite family of equiangular tight frames with Heisenberg symmetry. (These are not SIC-POVMs, but they appear to be related.)