3:00pm - 5:00pm
Algebra, geometry, and combinatorics of subspace packings
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 fourth session is "Symplectic and real algebraic geometry in frame theory."
(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)
Symplectic Geometry and Frame Theory
Geometric tools are increasingly important in the study of (finite) frames, which are simply redundant bases that are useful in signal processing and other applications where robustness to noise and erasures are important. Symplectic geometry was developed as the right general setting for Hamiltonian mechanics and in practice is often quite closely related to complex geometry: for example, smooth projective varieties are always symplectic manifolds. This is a promising, though mostly unexplored, collection of tools to be applied to the theory of frames in complex vector spaces.
The slogan is that any property which can be characterized as a level set of a moment map is likely to be amenable to symplectic techniques. In particular, unit-norm tight frames – which are particularly useful for applications – arise as the level set of a natural Hamiltonian group action on the set of complex matrices of a given size.
In this talk I will describe how symplectic tools can be used to generalize the frame homotopy theorem of Cahill–Mixon–Strawn and to give new insight into the Paulsen problem.
Symplectic Geometry, Optimization and Applications to Frame Theory
In recent work with Clayton Shonkwiler, we show that any space of complex frames (considered up to global rotations) with a prescribed frame operator can naturally be endowed with an extra geometric structure called a symplectic form. The goal of this talk is to explain how classical results from symplectic geometry can be used to provide theoretical guarantees for the convergence of optimization algorithms arising in frame theory. More specifically, spaces of frames with prescribed frame operator admit torus actions which are compatible with the symplectic structure (the torus actions are Hamiltonian). A result of Duistermaat says that gradient flows of certain functionals associated to Hamiltonian actions have no spurious local minima. We will discuss applications of this framework to the Paulsen problem from frame theory.
The optimal packing of eight points in the real projective plane
How can we arrange $n$ lines through the origin in three-dimensional Euclidean space in order to maximize the minimum angle between pairs of lines? Conway, Hardin and Sloane (1996) produced numerical line packings for $n leq 55$ that they conjectured to be optimal in this sense, but until now only the cases $n leq 7$ have been solved. We will discuss the resolution, joint with Dustin Mixon, of the case $n = 8$. Drawing inspiration from recent work on the Tammes problem, we proceed by enumerating potential contact graphs for an optimal configuration and eliminating those that violate various combinatorial and geometric constraints. The contact graph of the putatively optimal numerical packing of Conway, Hardin and Sloane is the only graph that survives, and we convert this numerical packing to an exact packing through cylindrical algebraic decomposition. We will further describe some potential improvements to our approach that could yield more exact optimal packings.
Spherical configurations with few angles
Let X be a spherical code in d-dimensional space. The degree of X is the number of inner products <u,v> that occur as u and v range over pairs of distinct elements from X. We are interested in spherical codes of small degree that arise from, or give rise to, association schemes. We will discuss equiangular lines, real mutually unbiased bases, work of Kodalen on "linked simplices" and joint work with Kodalen on "orthogonal projective doubles". A set of k full-dimensional simplices on the unit sphere is said to be "linked" if only two possible angles occur between vectors in distinct simplices. Given a graph G with vertex set V, a set L of lines through the origin in d-dimensional space is an "orthogonal projective double" of G if there is a bijection V --> L that maps adjacent pairs of vertices to orthogonal pairs of lines and non-adjacent pairs to lines forming some fixed angle between zero and 90 degrees. There is one aspect of this study involving elementary algebraic geometry. The ideal of X is the set of polynomials in d variables that vanish on each point in X and our goal is to determine, for each of the families mentioned above, a generating set for this ideal consisting of polynomials all having lowest possible total degree. This talk is based in part on joint work with my student Brian Kodalen and is supported by the US National Science Foundation.