DMS Colloquium: Emily King

Speaker: Emily King (U. Bremen)

Title: (Hilbert Space) Frames, Algebraic Combinatorics, and Geometry

Abstract: Frames are generalizations of orthonormal bases which yield "nice" decompositions of data. Such systems are the foundation of applied harmonic analysis and are also closely related to quantum measurements and linear codes.  When one wants an optimally robust representation of data, one often looks for frames that have some sort of spread, be it geometric (as non-parallel as possible) or algebraic (no nontrivial linear dependencies).  Over the last few years, it has been discovered that the relationship between these two types of spread is more complicated than had previously been believed. Furthermore, methods from algebra, geometry, and combinatorics have recently proven themselves to be very useful in the study of frames.

For example, algebraic graph theory and combinatorial design theory have led to new characterizations and novel constructions of optimal line (or, more generally, subspace) configurations which are also frames. Also, almost all desirable classes of frames form real algebraic varieties, and certain known results in frame theory have also been found to be equivalent to concepts in matroid theory and arrangements of hyperplanes.

In this talk, the currently known connections between these objects from harmonic analysis / quantum information theory and combinatorics / algebraic & discrete geometry will be presented.

(photo courtesy Uni Bremen/Kai Uwe Bohn)