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Title:
Probability theory for random groups arising in number theory
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Abstract:
We discuss the probability theory, and in particular the moment problem and universality theorems, for random groups of the type that often arise in number theory, topology, and combinatorics. We will discuss what the moments of random groups are, and when random groups are determined and not determined by their moments, describing both general results and particular examples for random groups that naturally arise (e.g. Selmer groups of random elliptic curves). We also discuss universality theorems, in the sense of the Central Limit Theorem, that say that asymptotically, random groups built from many independent inputs do not depend on the distribution of the inputs, and applications such as to the probability that a random n×m integral matrix gives a surjective linear transformation from Zm→Zn, and to the distribution of sandpile groups of random graphs. This talk is based partly on joint with with Will Sawin and Hoi Nguyen.
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