In this talk we will discuss recent work on the divisibility by 3 of class numbers of quadratic fields, and the exponents of class groups of imaginary quadratic fields. It is conjectured that the 3-part of the class number of the quadratic field Q(sqrt{D}) may be bounded above by an arbitrarily small power of |D|. However, until recently, the only known bound was the trivial bound O(|D|^{1/2 + epsilon}). Bounding the 3-part can be reduced to the problem of counting the number of squares of the form 4x^3 - dz^2, where d is a square-free positive integer, and x and z lie in the ranges x << d^{1/2}, z<< d^{1/4}. This counting problem is nontrivial because of the disproportionate ranges of the variables. We show that using a variant of the square sieve and the q-analogue of van der Corput's method allows one to tackle such a counting problem successfully. As a result, we show that the 3-part of the class number of the quadratic field Q(sqrt{D}) may be bounded by O(D^{27/56 + epsilon}). This gives a corresponding bound for the number of elliptic curves over the rationals with fixed conductor. We will also discuss recent work by Heath-Brown using this same method of counting points to give an upper bound for the size of discriminants of imaginary quadratic fields whose class groups can have an exponent of 5.

In the second lecture we shall discuss some open questions concerning nonlinear problems in geometry and related fully nonlinear equations on manifolds.