Speaker:: Professor Timothy Scott Trudgian. Mathematical Sciences Institute. The Australian National University Venue: B1044 Time: 14:30-15:30, 16:00-17:00   May. 17 Title: Linnik’s approximation to Goldbach’s conjecture Abstract: Goldbach’s conjecture is that 2N = p + q, for primes p and q, and for N sufficiently large. This is ‘almost’ true in that one needs to add at most finitely many powers of 2 to p+q to obtain 2N. This is Linnik’s theorem. I shall describe some of the ingredients of modern versions of Linnik’s theorem, and outline some potential improvements. Title: On Diophantine quintuples Abstract:   Consider the set {1,3,8}: the product of any two of its elements is one less than a perfect square. It is conjectured that there are no sets of five natural numbers with this property. I shall describe the history of this problem, the current state of knowledge, and some possible paths to a solution.