Clay Shonkwiler (CSU): "Symplectic Geometry and Frame Theory"
- Friday, November 9, 2018 from 3:00pm to 4:00pm
- Wilson Hall, 1-144 - view map
Speaking loosely, a frame for a (real or complex) Hilbert space is a spanning set, a tight frame is one which satisfies a version of Parseval's identity, and a unit-norm frame is one in which each of the vectors comprising the frame is a unit vector. A finite unit-norm tight frame (FUNTF) is a unit-norm tight frame on a finite-dimensional vector space. These generalizations of orthonormal bases are provably optimal for reconstructing signals in the presence of certain forms of noise and data loss, and have important applications in signal processing.
While frame theory has its roots in harmonic analysis, many developments in finite frames in the last decade or so have been driven by connections to and techniques from algebraic geometry. In this talk I will describe a promising new connection to symplectic geometry which is even more recent. In this setting, spaces of frames with desirable properties arise a level sets of moment maps associated to certain Hamiltonian group actions. In particular, I will give a simple proof of the Frame Homotopy Conjecture, which asserts that the space of complex FUNTFs of a given size in a given dimension is connected. This symplectic approach both avoids the tangle of inequalities that went into Cahill, Mixon, and Strawn's 2017 proof of the Frame Homotopy Conjecture and generalizes the result to a much broader class of frames. This is joint work with Tom Needham (Ohio State).
- david ayala
435 216 8073