Rational points via the circle method

Tim Browning

I will survey recent highlights around applications of the Hardy-Littlewood circle method to rational points, before focusing on what a version of this method over global fields of positive characteristic has to say about the geometry of rational curves on smooth hypersurfaces of low degree. Inspired by recent ideas of Peyre, moreover, I will also describe the parallel situation over the rational numbers, in which one counts points of bounded height satisfying the additional constraint that an associated tangent lattice is not too lopsided.