How big, typically, is the largest prime factor of an integer between n and 2n? This and many other related questions have answers with a strong probabilistic flavour, for reasons most completely formalized by Kubilius (1956, 1964); this has led to the development of a fruitful interaction between probability and number theory.

How big, typically, is the number of individuals in the largest allelic class in a population of size N, evolving under selectively neutral mutation? The answer (a Monash theorem) is given by the celebrated Ewens Sampling Formula, which reveals an astonishingly precise algebraic counterpart to the number theoretic setting above.

In this talk, we explain the simple, yet very general, probabilistic structure underlying this algebraic version, and show how it can be used to approach Knopfmacher's (1979) algebraic arithmetic. This is joint work with R. Arratia, S. Tavare.
 

Convenors:Kais Hamza, John Lattanzio, Marty Ross, Simon Clarke