The question of “levels of selection” has received a great deal of attention and controversy in the biological literature but relatively little attention in the mathematical literature. In this lecture we brie y review some background and formulate some models of multitype populations undergoing mutation and selection including multilevel selection. The main tool is a related dual processes whi...

We present a general formalism for giving a measure space paired with a separable Hilbert space a quantum version based on a normalized positive operator-valued measure. The latter are built from families of density operators labeled by points of the measure space. We especially focus on various probabilistic aspects of these constructions. Simple or more elaborate examples illustrate the proce...