The moment problem for groups and beyond

When is the distribution of a random variable determined by its moments? For real-valued random variables, this is the content of the classical moment problem. Similar problems exists for random groups. These arose in number theory in the course of understanding the behavior of class groups. In joint work with Melanie Matchett Wood, we give a resolution to the moment problem that applies uniformly to random groups, random natural numbers, and random objects in many other categories like random modules and random rings. Our methods have applications in number theory, to the distribution of class groups, and in topology, to the fundamental groups of random 3-manifolds.