School of Mathematics
"Beyond endoscopy", broadly interpreted, is the idea that functoriality should be realized as a comparison between stable trace formulas. The nature of this comparison, however, remains completely unclear.
Arithmetic complexity is considered (for many good reasons) simpler to understand than Boolean complexity. And indeed, we seem to have significantly more lower bound techniques and results in arithmetic complexity than in Boolean complexity. Despite many successes and rapid progress, however, foundational challenges, like proving super-polynomial lower bounds on circuit or formula size for explicit polynomials, or super-linear lower bounds on explicit 3-dimensional tensors, remain elusive.