Codes and Expansions (CodEx) Seminar
Tom Needham (Florida State University)
Applications of Symplectic Geometry to Frame Theory
Symplectic geometry was originally introduced as a mathematical formalism for classical mechanics and has since developed into its own rich subfield of differential geometry. It is a useful tool for studying certain Lie group actions on complex vector spaces, and ideas from symplectic geometry therefore apply very naturally in the setting of complex frames. This perspective gives new proofs of existing results on the geometry and topology of spaces of finite unit norm tight frames (FUNTFs) and allows one to extend these results to more general spaces of frames with prescribed frame norms and frame operators. Properties of frame spaces which are observable from the symplectic point of view include connectivity (frame homotopy), density of full spark frames and existence of singular points. In this talk, I will describe key concepts from symplectic geometry, illustrate them in the concrete setting of frames, and show how they are applied to get new results in frame theory. This is joint work with Clayton Shonkwiler.