As old and ubiquitous positive definite Hermitian forms may be, they still provide surprises and unexpected ramifications to other fields. A theorem of Quillen about positive definite forms on an odd dimensional sphere will be the starting point of our lecture, leading to a novel classification of real algebraic varieties and a precise Positivstellensatz in the Hermitian context. Applications to the construction of tight wavelet frames and to global, non-convex optimization will be presented.