A complete classification of the zero-dimensional homogeneous spaces under determinacy

All spaces are assumed to be separable and metrizable. We give a complete classification of the zero-dimensional homogeneous spaces, under the Axiom of Determinacy. This classification is expressed in terms of topological complexity (in the sense of Wadge theory) and Baire category. In the same spirit, we also give a complete classification of the filters on $\omega$ up to homeomorphism. As byproducts, we obtain purely topological characterizations of the semifilters and filters on $\omega$. The Borel versions of these results are in almost all cases due to Fons van Engelen. Along the way, we obtain Wadge-theoretic results of independent interest, especially regarding closure properties.