It is a standard heuristic that sums of oscillating number theoretic functions, like the M\"obius function or Dirichlet characters, should exhibit squareroot cancellation. It is often very difficult to prove anything as strong as that, and we generally expect that if we could prove squareroot cancellation it would be the best possible bound. I will discuss recent results showing that, in fact, certain averages of multiplicative functions exhibit a bit more than squareroot cancellation

Let V be an infinite vector space over a finite field k and let be an approximate homomorphism from V to End(V), i.e. the rank of f(x+y)-f(x)-f(y) is uniformly bounded by some constant r. We show that there is a homomorphism g such that f-g is of rank uniformly bounded by R(r).

We introduce a notion of a approximate cohomology groups and interpret this result as a computation of H^1 in a special case. Our proof uses the inverse theorem for the Gowers norms over finite fields; and an independent proof of this result (and some natural generalizations of it) would lead to a new proof of the inverse theorem. Joint work with D. Kazhdan

Arithmetic quantum chaos concerns the limiting behavior of a sequence of automorphic forms with parameters tending off to infinity. It is now known in many cases that the mass distributions of such forms equidistribute. Unfortunately, the known rates of equidistribution are typically weak (ineffective or logarithmic). I will discuss the problem of obtaining strong rates (power savings) and the related subconvexity problem, emphasizing recent progress concerning the level aspect on hyperbolic surfaces.

When we consider the average size of the p-torsion in class groups of quadratic fields, the behavior for p=2 is controlled by genus theory and is different from the conjectured behavior for odd primes, which is uniform in a certain sense over all odd primes. In this talk, we will consider the question when quadratic fields are replaced by fields of higher degree--for which primes p are the class group averages "bad" versus "good"? We will explain some theorems on class group averages for extensions of function fields that show some subtleties in the classification of good and bad primes

I will discuss joint work with M. Radziwill and T. Tao on asymptotics for the sums $\sum_{n \leq x} \Lambda(n) \Lambda(n+h)$ and $\sum_{n \leq x} d_k(n) d_l(n+h)$ where $\Lambda$ is the von Mangoldt function and $d_k$ is the kth divisor function. For the first sum we show that the expected asymptotics hold for almost all $|h| \leq X^{8/33}$ and for the second sum we show that the expected asymptotics hold for almost all $|h| \leq (\log X)^{O_{k, l} ( 1) }$.

We will describe recent progress in our ongoing program with Jean Bourgain to understand a number of different problems through the lens of thin orbits. An important role will be played by the production of levels of distribution (in certain Affine Sieves) which go "beyond expansion." No prior knowledge of these topics is assumed

I will discuss recent joint work on the value distribution of arithmetic functions. The problems discussed have in common that they owe their origin --- in one way or another --- to the fascination of the ancients with sums of divisors

One of the most spectacular results on arithmetic of Apollonian circle packings is the "almost" local to global principle for curvatures in any given integral Apollonian packing as described by Bourgain-Kontorovich in 2014. The methods in their work, inspired originally by an observation of Sarnak’s in his letter to Lagarias on Apollonian circle packings, apply to a much larger class of circle packings. In this talk, we clarify what "almost" local to global means, and describe what the larger class is, as well as what aspects of the packings in this class seem necessary in order to conclude an "almost" local to global result and how they enter the proof. This is joint work with Stange and Zhang.

Part-and-parcel of the study of "multiplicative number theory" is the study of the distribution of multiplicative functions in arithmetic progressions. In this talk I will discuss a Bombieri-Vinogradov type theorem for multiplicative functions and develop some limitations. This is joint work with Andrew Granville.