Talk

By a Theorem of Cobham, it is known that the support of an automatic sequence up to N grows either polylogarithmically in N or at least as a fractional power of N. We will illustrate a new proof of this that uses graph-theoretic tools. To do this we introduce the notion of a cycle arborescence and its height and we will explain where this height shows up in the asymptotic formula for both cases.

We show that the modern version of the circle method powered by the equidistribution of quadratic roots allows us to count special points on quadratic surfaces. For example, we obtain asymptotics for integer points on quadratic surfaces with prime coordinates and in short intervals.

The elliptic gamma function was (re)discovered in 1997 by Ruijsenaars; it gives rise to solutions of models in statistical / mathematical physics. It has a surprising number-theoretic appearance as demonstrated recently by Bergeron, Charollois and García. In parallel with the role of classical Jacobi theta form describing the class field for quadratic imaginary fields, the elliptic gamma function (conjecturally) does this task for cubic fields with a complex embedding.

By a result of Milnor we know that every spin 3-manifold spin bounds a 4-manifold. However, if we require this null-bordism to be of a prescribed normal 1-type, then it may no longer exist. In the spin case this normal $1$-type is a spin structure on the 3 manifold Y, together with the map on fundamental groups $\pi_1(Y) \to \pi_1(X)$ that we wish to realise via a bounding $4$-manifold $X$. We describe a three-stage geometric obstruction theory for the existence of a filling which extends this structure.