Frame theory intersects geometry

This workshop, sponsored by AIM and the NSF, will be devoted to outstanding problems that are in the intersection of frame theory and geometry. Frames are families of vectors in Hilbert spaces which provide stable expansions. These families are more general than orthonormal bases because they can incorporate linear dependencies. Applications in engineering, quantum theory and in pure mathematics have lead to several design problems for which optimal frames satisfy spectral as well as geometric properties. The simplest frame design problem, the construction of equal-norm Parseval frames, can be rephrased as a minimization problem on a Stiefel manifold. Similar reformulations exist for the construction of equiangular tight frames, including the special case of symmetric informationally complete positive operator-valued measures in quantum information theory. Another design problem with geometric character is that of frames for phase retrieval; these frames allow the recovery of a point in complex projective space based on the magnitudes of its frame coefficients.

Recent advances on these types of problems have incorporated more and more geometric techniques in their analysis. A strong interaction between researchers in frame theory with those in real and complex geometry, algebraic geometry and algebraic topology is expected to boost progress on these outstanding problems.

Particular topics envisioned for the workshop are the following:

1. Existence of complex equiangular line sets and equiangular Parseval frames as affine algebraic varieties. The construction of equiangular Parseval frames amounts to solving a number of polynomial equations defining a possibly non-trivial algebraic variety. The existence of such varieties could be shown by constructing Gröbner bases for the associated ideal space.
2. Connectedness of equal-norm Parseval frames and convergence of gradient flows towards global minimizers. The connectedness is a necessary condition for the feasibility of constructing Grassmannian frames by the minimization of suitable frame potentials. Convergence results are expected from local convexity.
3. Characterization of frames which allow phase retrieval from magnitudes of frame coefficients. The construction of such frames is equivalent to having uniqueness of solutions for certain quadratic equations in projective space.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

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