I will discuss moments of partially smoothed, truncated divisor sums of the M\"obius function. Such divisor sums appear naturally in the theory of the Selberg sieve and they play a key role in the GPY sieve and its recent improvements due to Maynard and Tao. It turns out that if the truncation is smooth enough, the main contribution to the moments comes from almost primes. However, for rougher truncations, the dominant contribution comes from integers with many prime factors. Analogous questions can be asked for polynomials over finite fields and for permutations, and in these cases the moments behave rather differently, a rare exception. As we will see, a plausible explanation for this phenomenon is given by studying the analogous sums for Dirichlet characters and obtaining different answers depending on whether or not the character is ``exceptional''. This is joint work with Andrew Granville and James Maynard

Let K denote a number field of degree n, and for a fixed, positive integer l, consider the l-torsion subgroup of the class group of K. It is conjectured that the size of the this l-torsion subgroup is very small (in an appropriate sense), relative to the absolute discriminant of the field K. In 2007, Ellenberg and Venkatesh proved a nontrivial bound (removing a power from the trivial bound) by assuming GRH. In this talk, we will discuss a method that recovers this bound for almost all members of certain families of fields, without assuming GRH. This is joint work with Lillian Pierce and Melanie Matchett Wood

The last years have seen a series of breakthroughs in the understanding of mean values related to exponential sums, that give rise to new improved estimates on the number of solutions to diagonal equations. This includes the proof of Vinogradov's Mean Value Theorem by Wooley and Bourgain, Demeter and Guth, as well as close-to-perfect mean value estimates for systems of diagonal cubic equations due to Bruedern and Wooley.

I will give an account of recent progress regarding mixed systems consisting of both cubic and quadratic equations. Building on the methods of Wooley and Bruedern-Wooley, we establish asymptotic estimates for the number of solutions of such systems, provided that the number of variables is not much larger than what is required by square root cancellation, and in a few cases we achieve bounds of this quality. This work is partly joint with Scott Parsell