10:00am - 12:00pmAlgebra, geometry, and combinatorics of subspace packings
Chair(s): Emily Jeannette King (University of Bremen, Germany), Dustin Mixon (Ohio State University)
Frame theory studies special vector arrangements which arise in numerous signal processing applications. Over the last decade, the need for frame-theoretic research has grown alongside the emergence of new methods in signal processing. Modern advances in frame theory involve techniques from algebraic geometry, semidefinite programming, algebraic and geometric combinatorics, and representation theory. This minisymposium will explore a multitude of these algebraic, geometric, and combinatorial developments in frame theory.
The theme of the first session is "Systems with non-abelian group symmetry."
(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)
Algebra, Geometry, and Combinatorics of Subspace Packings: Gabor-Steiner Equiangular Tight Frames
Emily King
University of Bremen
Desirable traits of subspace arrangements in applications like signal processing and quantum information theory include having large geometric spread between any two subspaces and yielding a resolution of the identity. Methods from algebraic graph theory, real algebraic geometry, symplectic geometry, combinatorial design theory, semidefinite programming, and more can be used to design and characterize such subspace packings. This talk will serve as an introduction to the minisymposium “Algebra, Geometry, and Combinatorics of Subspace Packings.” Gabor-Steiner equiangular tight frames, which are covariant under the Weyl-Heisenberg group and have properties described by different combinatorial designs will also be discussed.
Group frames, full spark, and other topics
Romanos-Diogenes Malikiosis
Aristotle University of Thessaloniki
A group frame is the orbit of a vector in a vector space of dimension N under the action of a (projective) linear representation of a finite group. Such a frame satisfies the full spark property, if every selection of N vectors from the frame constitutes a basis.
We will examine whether certain families of group frames satisfy the full spark property, extending the results by the speaker for the Weyl-Heisenberg group (i.e. Gabor frames) and by Oussa-Sheehan for the dihedral case. If time allows, we will also mention other topics as well, for example equiangularity.
This is joint work with Vignon Oussa.
Equiangular tight frames from nonabeilan groups
John Jasper
South Dakota State University
Several applications in signal processing require lines through the origin of a finite-dimensional Hilbert space with the property that the smallest interior angle is as large as possible. Packings that achieve equality in the Welch bound are known as equiangular tight frames (ETFs). Since optimal packings often exhibit symmetry, it is natural to expect such packings to be related to groups. Indeed, a popular type of ETFs are the so-called harmonic ETFs, that is, ETFs that arise from the action of an abelian group on a single vector. On the other hand, perhaps the most famous open problem in this area is Zauner's conjecture, which asks for an ETF from the action of the Heisenberg group, which is nonabelian. The theory of harmonic ETFs is fairly well understood as it is equivalent to well-studied objects known as difference sets. The theory of ETFs generated by nonabelian groups is much more mysterious. In this talk we will discuss this theory and present a construction of the first infinite family of ETFs arising from nonabelian groups.
SIC-POVM existence and the Stark conjectures
Gene Kopp
University of Bristol
The existence of a configuration of equiangular lines in d-dimensional complex Hilbert space of cardinality achieving the theoretical upper bound of d^2 is known only for finitely many dimensions d. Such configurations have been studied extensively in the context of quantum information theory, in which they are known as symmetric informationally complete positive operator-valued measures (SIC-POVMs).
We give an explicit conjectural construction of SIC-POVMs in an infinite family of dimensions. Our construction uses values of derivatives of zeta functions at s=0 and is closely connected to the Stark conjectures over real quadratic fields. Moreover, in the same family, we prove a conditional result stating that SIC-POVMs exist under a strong algebraic hypothesis about units in a certain number field. The talk will include a worked example in dimension d=5 and an overview of some number-theoretic background necessary to understand the main results.