Cubic surfaces in affine three space tend to have few integral points .However certain cubics such as $x^3 + y^3 + z^3 = m$, may have many such points but very little is known. We discuss these questions for Markoff type surfaces: $x^2 +y^2 +z^2 -x\cdot y\cdot z = m$ for which a (nonlinear) descent allows for a study. Specifically that of a Hasse Principle and strong approximation, together with "class numbers" and their averages for the corresponding nonlinear group of morphims of affine three space.[-]

The Chowla conjecture asserts that the signs of the Liouville function are distributed randomly on the integers. Reinterpreted in the language of ergodic theory this conjecture asserts that the Liouville dynamical system is a Bernoulli system. We prove that ergodicity of the Liouville system implies the Chowla conjecture. Our argument has an ergodic flavor and combines recent results in analytic number theory, finitistic and infinitary decomposition results involving uniformity norms, and equidistribution results on nilmanifolds.[-]

We will discuss a new type of ergodic theorem which has among its corollaries numerous classical results from multiplicative number theory, including the Prime Number Theorem, a theorem of Pillai-Selberg and a theorem of Erdös-Delange. This ergodic approach leads to a new dynamical framework for a general form of Sarnak's Möbius disjointness conjecture which focuses on the "joint independence" of actions of (N,+) and (N,x). The talk is based on recent joint work with Florian Richter.[-]

Roots of unity of order dividing $n$ equidistribute around the unit circle as $n$ tends to infinity. With some extraeffort the same can be shown when restricting to roots of unity of exact order $n$. Equidistribution is measured by comparing the average of a continuous test function evaluated at these roots of unity with the integral over the complex unit circle. Baker, Ih, and Rumely extended this to test function with logarithmic singularities of the form $\log|P|$ where $P$ is a univariate polynomial in algebraic coefficients. I will discuss joint work with Vesselin Dimitrov where we allow $P$ to come from a class of a multivariate polynomials, extending a result of Lind, Schmidt, and Verbitskiy. Our method draws from earlier work of Duke.[-]