Abstract: I will introduce the notion of a "parametrisable variety" which encapsulates the idea of being able to parametrise the rational points on a variety. I will use this to give some applications to Noether's approach to the inverse Galois problem.

Abstract: I will report on joint work in progress with Ross Paterson and Happy Uppal in which we explore the Hasse principle for zero-cycles on diagonal surfaces of prime degree. Our approach goes via the descent-fibration method, which Swinnerton-Dyer used to treat the degree-3 case.

Abstract: TBC

Abstract: Let K be a p-adic field, i.e. a finite extension of the p-adic numbers Qp. Artin conjectured that if n is at least d^2, every projective hypersurface in P^n of degree d has a K-point. While this turns out to be false in general (there is a quartic counterexample over Q_2), the conjecture holds for quadrics and cubics due to Hasse and Lewis, respectively; moreover, it remains open for prime degrees d. In this talk, we describe recent progress in degrees 5, 7, and 11 to give effective bounds on the size of the residue field for which the conjecture is known to hold. A wide range of techniques are needed, including effective Bertini theorems, point counting over finite fields, and computation.

Abstract: TBC

Abstract: TBC

Abstract: Malle's conjecture gives a prediction for the number of A_4 quartic number fields with bounded discriminant.  I will discuss recent work with Daniel Loughran, where we give a lower bound for this quantity which matches Malle's prediction.  I'll explain how we did this, and in the process convince you that this is a talk about rational points!