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:  I will report on work in progress with Francesca Balestrieri, Anouk Greven, Soumya Sankar, Katerina Santicola and Manoy Trip in which we describe an obstruction built from Brauer—Manin and descent that is sufficient to explain all failures of the Hasse principle and weak approximation for 0-cycles on Kummer varieties, assuming finiteness of Tate—Shafarevich groups for the underlying abelian varieties.

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: In this talk I will present work in progress with Jessica Alessandrì, in which we study the Brauer group of affine surfaces given by "Fermat near misses" equation. They can be obtained by removing an elliptic curve from degree 2 del Pezzo surfaces. I will focus on specific examples and explain the methods we use to analyse the Brauer group of these open surfaces.

Abstract: An unlikely intersection arises when an algebraic subvariety meets another in a way that is not predicted by naive dimension counts. In Shimura varieties, the expectation that unlikely intersections with special subvarieties should be sparse is formalised by the Zilber–Pink conjecture, which unifies results such as Manin–Mumford, Mordell–Lang, and André–Oort. I will explain how comparing the various notions of complexity associated with special subvarieties helps underpin progress toward Zilber–Pink. 

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!