Let $\alpha, \beta$ be elements of infinite order in the circle group. A closed set K in the circle is called an \alpha \beta set if for every x\in K either x+\alpha \in K or x+\beta \in K. In 1979 Katznelson proved that there exist non-dense \alpha \beta sets, and that there exist \alpha \beta sets of arbitrarily small Hausdorff dimension. We shall discuss this result, and a more recent result of Feng and Xiong, showing that the lower box dimension of every \alpha \beta set is at least 1/2.
An old result of Hedlund tells us that there are no closed orbits for the horocycle flow on a compact Riemann surface M. The situation is different if M is non-compact in which case there is a one-parameter family of periodic orbits for every cusp of M. I want to talk about a result by Sarnak concerning the distribution of the such orbits in each of these families when their length goes to infinity. It turns out that these orbits become equidistributed in M and the rate of convergence can in fact be quantified in terms of spectral properties of the Eisenstein series on M.
A classical theorem of Erdos and Turan states that if a monic polynomial has small values on the unit circle (relative to its constant coefficient), then its zeros cluster near the unit circle and are close to being equidistributed in angle. In February 2018, K. Soundararajan gave a short and elementary proof of their result using Fourier analysis. I'll present this new proof.
The Mass Transport Principle is a useful technique that was introduced to the study of automorphism-invariant percolations by Häggström in 1997. The technique is a sort of mass conservation principle, that allows us to relate random properties (such as the random degree of a vertex) to geometric properties of the graph.
I will introduce the principle and the class of unimodular graphs on which it holds, as well as a few of its applications.
One of the first algorithm any mathematician learns about is the Euclidean division algorithm for the rational integer ring Z. When asking whether other integer rings have similar such division algorithms, we are then led naturally to a geometric interpretation of this algorithm which concerns lattices and their (multiplicative) covering radius.