This project concerns rational points, which are essentially whole-number solutions to equations, a fundamental object of interest in mathematics that has been studied since ancient times but is even now understood only to a very limited extent. The modern point of view is that the equations define a geometric object, called a variety, such as a curve or a surface.

To give a basic example, consider the equation $x^2+y^2=1$, which describes the unit circle in the sense that the points $(x , y)$ in the xy -plane that satisfy this equation make up this circle. The points $(4/5, 3/5)$ and $(5/13, 12/13)$ on this circle have coordinates that are fractions, also called rational numbers; these points are called rational points. To Pythagoras, and even in ancient Babylonia and India, it was known that this circle contains infinitely many such rational points, though they did not phrase it in these geometric terms. In contrast to the case of the circle, Fermat proved in the 17th century that the only rational points on the curve given by $x^4 + y^4 = 1$ are $(\pm1, 0)$ and $(0, \pm1)$.

By viewing whole-number solutions as rational points on the variety described by the equations, we open the door to tools from algebraic geometry. Since the 20th century, this geometric point of view has led to major advances. Moreover, it turns out that the geometry of a variety determines to an amazing extent the qualitative and quantitative behaviour of its rational points.

In the case of curves, which have dimension 1, we can associate with each curve a geometric invariant, called its genus; a celebrated theorem by Faltings (1983) states that if the genus is at least 2, then there are only finitely many rational points on the curve. This is indeed consistent with the examples above, as the circle has genus 0, while the second curve has genus 3.

For varieties of higher dimension, it is generally believed that their geometry still governs the distribution of rational points. The geometry, however, is much more subtle. There are long-standing conjectures, such as those by Colliot-Thélène (1980), Batyrev–Manin (1989) and Bombieri–Lang (1974), that formulate in a precise way how the behaviour of rational points is affected by the geometry. The study of rational points, particularly on surfaces and higher-dimensional objects, has flourished in recent years.