# Conditional approaches to sums of cubes

# Conditional approaches to sums of cubes

**In-Person and Online Talk **

**Zoom link: ** **https://princeton.zoom.us/j/97126136441**

**Passcode : T**

**he three digit integer that is the cube of the sum of its digits**

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 x_1^3+...+x_6^3 = 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.