Strucure and Randomness in the Prime Numbers

"God may not play dice with the universe, but something strange is going on with the prime numbers" - Paul Erdos The prime numbers are a fascinating blend of both structure and randomness. It is widely believed that beyond the ‘obvious' structures in the primes, they otherwise behave as if they were distributed randomly; this ‘pseudorandomness' then underlies our belief in many unsolved conjectures about the primes, from the twin prime conjecture to the Riemann hypothesis. This pseudorandomness has been frustratingly elusive to actually prove rigorously, but recently there has been progress to establish new results about the primes, such as that they contain arbitrarily long arithmetic progressions. Some of these developments will be discussed in this lecture.