at the

American Institute of Mathematics, San Jose, California

organized by

Bernhard Bodmann, Gitta Kutyniok, and Tim Roemer

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:

- 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 Gr�bner bases for the associated ideal space.
- 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.
- 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 schedule.

A report on the workshop activities.

A list of open problems.

Papers arising from the workshop:

Connectivity and Irreducibility of Algebraic Varieties of Finite Unit Norm Tight Frames

by Jameson Cahill, Dustin G. Mixon and Nate Strawn, *SIAM J. Appl. Algebra Geom. 1 (2017), no. 1, 38–72 * MR3633768

Complex Two-Graphs via Equiangular Tight Frames

by Thomas Hoffman and James Solazzo

An algebraic characterization of injectivity in phase retrieval

by Aldo Conca, Dan Edidin, Milena Hering and Cynthia Vinzant, *Appl. Comput. Harmon. Anal. 38 (2015), no. 2, 346–356 * MR3303679

Improved Recovery Guarantees for Phase Retrieval from Coded Diffraction Patterns

by David Gross, Felix Krahmer and Richard Kueng, *Appl. Comput. Harmon. Anal. 42 (2017), no. 1, 37-64 * MR3574560

A strong restricted isometry property, with an application to phaseless compressed sensing

by Vladislav Voroninski and Zhiqiang Xu, *Appl. Comput. Harmon. Anal. 40 (2016), no. 2, 386-395 * MR3440178

All complex equiangular tight frames in dimension 3

by Ferenc Szöllősi

Frame potentials and the geometry of frames

by Bernhard G. Bodmann and John Haas, *J. Fourier Anal. Appl. 21 (2015), no. 6, 1344-1383 * MR3421919