I will describe some new-ish results on the average and maximum size of the Riemann zeta function in a "typical" interval of length 1 on the critical line. A (hopefully) interesting feature of the proofs is that they reduce the problem for the zeta function to an analogous problem for a random model, which can then be solved using various probabilistic techniques.

We will outline the main features of Wooley's efficient congruencing method for the parabola. Then we will go on to prove new bounds for discrete restriction to the curve (x,x^3). The latter is joint work with Trevor Wooley (Purdue).

The famous Birch & Swinnerton-Dyer conjecture predicts that the (algebraic) rank of an elliptic curve is equal to the so-called analytic rank, which is the order of vanishing of the associated L-functions at the central point. In this talk, we shall discuss the analytic rank of automorphic L-functions in an "alternate universe". This is joint work with Kyle Pratt and Alexandru Zaharescu.

We consider an analogue of Kontsevich's matrix Airy function where the cubic potential $\mathrm{Tr}(\Phi^3)$ is replaced by a quartic term $\mathrm{Tr}(\Phi^4)$. By methods from quantum field theory we show that also the quartic case is exactly solvable. All cumulants can be expressed as composition of elementary functions with the inverse of another elementary function. For infinite matrices the inversion gives rise to hyperlogarithms and zeta values as familiar from quantum field theory. For finite matrices the elementary functions are rational and should be viewed as branched covers of Riemann surfaces, in striking analogy with the topological recursion of the Kontsevich model. This rationality is strong support for the conjecture that the quartic analogue of the Kontsevich model is integrable.
 

We construct a functor from arithmetic schemes (and dominant morphisms) to dynamical systems which allows to recover the Hasse-Weil zeta function of a scheme as a Ruelle type zeta function of the corresponding dynamical system. We state some further properties of this correspondence and explain the relation to the work of Kucharczyk and Scholze who realize the Galois groups of fields containing all roots of unity as (etale) fundamental groups of certain topological spaces. We also explain the main reason why our dynamical systems are not yet the right ones and in what regard they need to be refined.
 

One can study periods of algebraic varieties by a process of "fibering out" in which the variety is fibred over a punctured curve $f:X->U$. I will explain this process and how it leads to the classical Picard Fuchs (or Gauss-Manin) differential equations. Periods are computed by integrating solutions of Picard Fuchs over suitable closed paths on $U$. One can also couple (i.e.tensor) the Picard Fuchs connection to given connections on $U$. For example, $t^s$ with $t$ a unit on $U$ and $s$ a parameter is a solution of the connection on $\mathscr{O}_U$ given by $\nabla(1) = sdt/t$. Our "periods" become integrals over suitable closed chains on $U$ of $f(t)t^sdt/t$. Golyshev called the resulting functions of $s$ "motivic Gamma functions". 
Golyshev and Zagier studied certain special Picard Fuchs equations for their proof of the Gamma conjecture in mirror symmetry in the case of Picard rank 1. They write down a generating series, the Apéry series, the knowledge of the first few terms of which implied the gamma conjecture. We show their Apéry series is the Taylor series of a product of the motivic Gamma function times an elementary function of $s$. In particular, the coefficients of the Apéry series are periods up to inverting $2\pi i$. We relate these periods to periods of the limiting mixed Hodge structure at a point of maximal unipotent monodromy. This is joint work with M. Vlasenko. 
 

Some of the simplest expected cases of Langlands functoriality are the symmetric power liftings Sym^r from automorphic representations of GL(2) to automorphic representations of GL(r+1). I will discuss some joint work with Jack Thorne on the symmetric power lifting for holomorphic modular forms.

Let E be an elliptic curve over the rationals and p a prime such that E admits a rational p-isogeny satisfying some assumptions. In a joint work with J. Lee and C. Skinner, we prove the anticyclotomic Iwasawa main conjecture for E/K for some suitable quadratic imaginary field K. I will explain our strategy and how this, combined with complex and p-adic Gross-Zagier formulae, allows us to prove that if E has rank one, then the p-part of the Birch and Swinnerton-Dyer formula for E/Q holds true.