About:

The topic of rational points on varieties over the rational numbers is the modern perspective on the theory of Diophantine equations.

There is a good (partially conjectural) understanding now of the situation for algebraic curves. The proof of the Mordell conjecture for curves of genus at least 2 by Faltings is one of the crowning achievements in the area, and much of the work on elliptic curves is driven by the Birch-Swinnerton-Dyer conjecture. Recent highlights include the work of Bhargava and his collaborators on average ranks of elliptic curves. The situation in higher dimensions is much murkier however.

The aim of the meeting was to bring together leading experts and early career researchers to make progress on understanding rational points on surfaces and higher dimensional varieties. Traditionally there have been two separate communities working in the area, using tools from analytic number theory and algebraic geometry, respectively. Spectacular progress has been made in recent times by managing to bridge these communities, with a particular highlight being applications of Green-Tao-Ziegler's work on primes in arithmetic progressions to the fibration method. The emphasis in the meeting was on building upon this bridge and further inspiring collaboration between the analytic and geometric communities.

Specific topics covered included the following:

1. Schinzel's Hypothesis with probability 1

2. Campana points

3. Purity of strong approximation

4. Rational points in families

5. Brauer--Manin obstruction