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, semidefiniteprogramming, 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 second session is "Equiangular lines."
(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)
Equiangular tight frames from group divisible designs
Matthew Fickus
Air Force Inst. of Technology
An equiangular tight frame (ETF) is a type of optimal packing of lines in a real or complex Hilbert space. In the complex case, the existence of an ETF of a given size remains an open problem for many choices of parameters. We discuss how many of the known constructions of ETFs are of one of two types. We further provide a new method for combining a given ETF of one of these two types with an appropriate group divisible design (GDD) in order to produce a larger ETF of the same type. By applying this method to known families of ETFs and GDDs, we obtain several new infinite families of ETFs.
Using Biangular Gabor Frames to Construct Equiangular Tight Frames
Mark Magsino
Ohio State University
Biangular Gabor Frames can be described by a system of real polynomial equations of many variables. Numerical computations suggests this system is usually one dimensional. Furthermore, a vector containing all ones leads to a biangular Gabor frame. This gives the idea of following the curve to an equiangular tight frame somewhere along this curve. We present numerical results of finding equiangular tight frames using this method.
Doubly transitive lines: Symmetry implies optimality
Joseph Iverson
Iowa State University
Since the work of Tóth on regular figures, it has been widely observed that optimal solutions to packing problems frequently display extraordinary symmetries. For instance, spheres centered on points in the Leech lattice give an optimal packing in 24 dimensions, while lines through through antipodal vertices of an icosahedron give an optimal packing in two-dimensional projective space. In this talk, we demonstrate an extreme case of this phenomenon for line packings: symmetry can be a sufficient condition for optimality. Specifically, consider n lines spanning a space of dimension d < n. If the lines have a doubly transitive automorphism group, then they are optimally packed in projective space. In fact, unit norm representatives for the lines reach equality in the Welch bound to create an equiangular tight frame. We will explain this phenomenon, and then discuss progress toward a classification of all doubly transitive lines.
This is joint work with Dustin G. Mixon.
Equiangular lines in $\mathbb R^{17}$ and the characteristic polynomial of a Seidel matrix
Gary Greaves
Nanyang Technological University
A system of lines through the origin of $mathbb R^d$ for which the angle between any pair of lines is a constant is called equiangular. A Seidel matrix, which can be interpreted as a variation of the adjacency matrix of a graph, is a tool for studying equiangular line systems. In this talk we present our recent improvement on the upper bound for the cardinality of an equiangular line system in $mathbb R^{17}$. A crucial ingredient for this improvement is a new restriction on the characteristic polynomial of a Seidel matrix.
This talk is based on joint work with Pavlo Yatsyna.