Oxford

Given a form $F(x)$, the circle method is frequently used to provide an asymptotic for the number of representations of a fixed integer $N$ by $F(x)$. However, it can also be used to prove results of a different flavor, such as showing that almost every number (in a certain sense) has at least one representation by $F(x)$. In joint work with Roger Heath-Brown, we have recently considered a 2-dimensional version of such a problem. Given two quadratic forms $Q_1$ and $Q_2$, we ask whether almost every integer (in a certain sense) is simultaneously represented by $Q_1$ and $Q_2$. Under a modest geometric assumption, we are able to prove such a result if the forms are in $5$ variables or more. In particular, we show that any two such quadratic forms must simultaneously attain prime values infinitely often. In this seminar, we will review the circle method, introduce the idea of a Kloosterman refinement, and investigate how such "almost all" results may be proved.

In the absence of background fluxes and sources, the compactification of string theories on Calabi-Yau threefolds leads to supersymmetric solutions.Turning on fluxes, e.g. to lift the moduli of the compactification, generically forces the geometry to break the Calabi-Yau conditions, and to satisfy, instead, the weaker condition of reduced structure. In this talk I will discuss manifolds with SU(3) structure, and their relevance for heterotic string compacitications. I will focus on domain wall solutions and how explicit example geometries can be constructed.

I will present an efficient algorithm for computing certain special values of Rankin triple product $p$-adic L-functions and give an application of this to the explicit construction of rational points on elliptic curves.