Anatomy of integers, polynomials and permutations

Abstract: There is a famous analogy between the statistics of the prime factors of a random integer, of the irreducible factors of a random polynomial over a finite field, and of the cycles of a random permutation. This analogy allows us to transfer techniques and intuition from one setup to the other, and it has been in the center of a lot of recent activity in probabilistic number theory and group theory. I will survey some of this progress, focusing in particular on results about the irreducibility of randomly chosen polynomials with 0,1 coefficients (joint with Lior Bary-Soroker and Gady Kozma), as well as on results about the concentration of divisors of random integers and the size of the Hooley Delta function (joint with Ben Green and Kevin Ford).

number theory

Audience: researchers in the topic

London number theory seminar

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For a record of talks predating this website see: wwwf.imperial.ac.uk/~buzzard/LNTS/numbtheo_past.html