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# Sublattices and subrings of Z^n and random finite abelian groups (Nathan Kaplan, UC Irvine)

## March 26 @ 12:15 pm - 1:10 pm

How many sublattices of **Z**^{n} have index at most X? If we choose such a lattice L at random, what is the probability that **Z**^{n}/L is cyclic? What is the probability that its order is odd? Now let R be a random subring of **Z**^{n}. What is the probability that **Z**^{n}/R is cyclic? We will see how these questions fit into the study of random groups in number theory and combinatorics. We will discuss connections to Cohen-Lenstra heuristics for class groups of number fields, sandpile groups of random graphs, and cokernels of random matrices over the integers.