Bounding the multiplicity of Fano toric varieties: ideas for a weight/topological classification

In this talk I will outline a roadmap to extend results of Conrads (2002) and Averkov-Kasprzyk-Lehmann-Nill (2021) on the multiplicity of fake weighted projective spaces to the case of higher Picard numbers, so obtaining some (non-sharp in dimension greater than or equal to 3) upper bounds for the multiplicity of a Fano toric variety, stratified on dimension and Picard number. As a byproduct and time permitting, I will sketch how to perform a topological classification on Fano toric varieties admitting a given weight matrix, starting from the crucial notion of the "weight group" of a Fano toric variety.