Matthew Buckland

Branching Interval Partition Diffusions

We construct an interval-partition-valued diffusion from a collection of excursions sampled from the excursion measure of a real-valued diffusion, and we use a spectrally positive Lévy process to order both these excursions and their start times. At any point in time, the interval partition generated is the concatenation of intervals where each excursion alive at that point contributes an interval of size given by its value. Previous work by Forman, Pal, Rizzolo and Winkel considers self-similar interval partition diffusions – and the key aim of this work is to generalise these results by dropping the self-similarity condition. The interval partition can be interpreted as an ordered collection of individuals (intervals) alive that have varying characteristics and generate new intervals during their finite lifetimes, and hence can be viewed as a class of Crump-Mode-Jagers-type processes.

Ofir Gorodetsky

Smooth and rough numbers

We all know and love prime numbers, but what about smooth and rough numbers?

We'll define y-smooth numbers -- numbers whose prime factors are all less than y. We'll explain their application in cryptography, specifically to factorization of integers.

We'll shed light on their density, which is modelled using a peculiar differential equation. This equation appears naturally in probability theory.

We'll also explain the dual notion to smooth numbers, that of y-rough numbers: numbers whose prime factors are all bigger than y, and in some sense generalize primes.

We'll explain their importance in sieve theory. Like smooth numbers, their density has interesting properties and will be surveyed.