In this talk, we discuss the reduction-quantization diagram in terms of formality. First, we propose a reduction scheme for multivector fields and multidifferential operators, phrased in terms of L-infinity morphisms. This requires the introduction of equivariant multivector fields and equivariant multidifferential operator complexes, which encode the information of the Hamiltonian action, i.e., a G-invariant Poisson structure allowing for a momentum map. As a second step, we discuss an equivariant version of the formality theorem, conjecturedby Tsygan and recently solved in a joint work with Nest, Schnitzer, and Tsygan. This result has immediate consequences in deformation quantization, since it allows for obtaining a quantum moment map from a classical momentum map with respect to a G-invariant Poisson structure.[-]

A finite unit norm tight frame (FUNTF) is a spanning set of unit vectors in a finite-dimensional Hilbert space such that the spectrum of singular values of an associated operator is constant. In signal processing applications, it is desirable to use FUNTFs to encode signals, as such representations are proven to be optimally robust to noise. This naturally gives rise to questions about the geometry and topology of the space of FUNTFs. For example, the conjecture that every space of FUNTFs is connected was open for 15 years, and slight variants of this problem still remain open. I will discuss recent work with Clayton Shonkwiler, where we answer several questions about random matrix theory and optimization in spaces of structured matrices, using tools from symplectic geometry and geometric invariant theory.[-]