Finite frames are simply finite spanning collections for real or complex d-dimensional space, but imposing some additional constraints on such collections yields frames that are ideal for wireless telecommunications, sigma-delta quantization, and coding theoretic applications. Some of the most studied structured frames are the finite unit-norm tight frames (FUNTFs), or the self-dual (up to a constant multiple) frames consisting of unit-length vectors. The frames with a fixed frame operator, and whose lengths are fixed generalizes the FUNTFs.

FUNTFs are characterized by algebraic constraints, and hence the space of FUNTFs is an algebraic variety. Based upon this observation, Ken Dykema initiated the geometric study of spaces FUNTFs and determined when this space is a manifold. In this series of talks, I shall present some results that probe the geometry of spaces of FUNTFs (and more general spaces of frames): I will show how nonsingular points and corresponding tangent spaces on frame spaces can be characterized, how the existence of structured local coordinate systems can be deduced, and how these coordinate systems on the spaces of frames can be explicitly constructed.

In this first talk, we shall demonstrate how distorted Stiefel manifolds and generalized tori intersect transversally in the Hilbert-Schmidt sphere. The remaining talks rely heavily upon the ideas presented in this talk.