Anatomy of integers and random permutations

This talk will address the questions "what is a random number?" and "what is a random permutation?" and make connections between the two seemingly dissimilar problems. I will describe how a certain Poisson model underlies both questions, from a factorization point of view, and how to use this to predict how the prime factors of typical integers and cycles of typical permutations are distributed. This has applications to interesting questions about divisors of integers, the familiar multiplication table, sets fixed by permutations, generation of the symmetric group,  and other questions.