Conditional approaches to sums of cubes
In 1986, Hooley applied (what practically amounts to) the general Langlands reciprocity (modularity) conjecture and GRH in a fresh new way, over certain families of cubic 3-folds. This eventually led to conditional near-optimal bounds for the number of integral solutions to x31+…+x36=0 in expanding boxes.
Building on Hooley's work, I will sketch new applications of large-sieve hypotheses, the Square-free Sieve Conjecture, and predictions of Random Matrix Theory type, over the same geometric families - e.g. conditional statistical results on sums of three integer cubes (a project suggested by Amit Ghosh and Peter Sarnak). These form the bulk of my thesis work (advised by Sarnak), and involve phenomena both random and structured, average- and worst-case, and multiplicative and additive.