In 1735, Euler observed that $ζ(2) = 1 + \frac{1}{2²} + \frac{1}{3²} + ⋯ = \frac{π²}{6}$. This is related to the famous identity $ζ(−1) "=" 1 + 2 + 3 + ⋯ "=" \frac{−1}{12}$. In general, values of the Riemann zeta function at positive even integers are equal to rational numbers multiplied by a power of $π$. The values at positive odd integers are much more mysterious; for example, Apéry proved that $ζ(3) = 1 + \frac{1}{2³} + \frac{1}{3³} + ⋯$ is irrational, but we still don't know if $ζ(5) = 1 + \frac{1}{2⁵} + \frac{1}{3⁵} + ⋯$ is rational or not! In this talk, we will explain the arithmetic significance of these values, their generalizations to Dirichlet/Dedekind L−functions, and to L−functions of elliptic curves. We will also present a new formula for $ζ(3) = 1 + \frac{1}{2³} + \frac{1}{3³} + ...$ in terms of higher algebraic cycles which came out of an ongoing project with Lambert A'Campo.

Khintchine's Theorem is one of the cornerstones in metric Diophantine approximation. The question of removing the monotonicity condition on the approximation function in Khintchine's Theorem led to the recently proved Duffin-Schaeffer conjecture. Gallagher showed an analogue of Khintchine's Theorem for multiplicative Diophantine approximation, again assuming monotonicity. In this talk, I will discuss my joint work with L. Frühwirth about a Duffin-Schaeffer version for Gallagher's Theorem. Furthermore, I will give a broader overview on various questions in metric Diophantine approximation and demonstrate the deep connection to both analytic and combinatorial number theory that is hidden inside the proof of these statements.

Functoriality is a key feature in Langlands’ conjectured relationship between automorphic representations and Galois representations; it predicts that certain Galois representations are automorphic, i.e. should come from automorphic representations. We discuss the idea of $p$-adic propagation of automorphy, which seeks to establish the automorphy of everything in a “neighborhood” given the automorphy of something in that neighborhood. The “neighborhoods” that we consider will be the irreducible components of a $p$-adic analytic space called the eigenvariety, which parameterizes $p$-adic automorphic representations. This technique was introduced by Newton and Thorne in their proof of symmetric power functoriality, and can be adapted to investigate similar problems.