The polynomial method is a general technique for capturing cancellations in spectral problems that has been developed in the past two years in several joint works with C.-F. Chen, J. Garza-Vargas, M. Magee, D. Puder, and J. Tropp. It has provided a powerful approach for proving optimal spectral gaps for random graphs, hyperbolic surfaces, and strong convergence problems. My aim in this talk is to describe in general terms how the method works, and what kind of spectral information it provides; in particular, I will highlight some applications beyond spectral gaps (based in part on recent work with Jorge Garza-Vargas).
