James Arthur

University of Toronto

October 1, 2016

Beyond Endoscopy is the proposal by Langlands for using the trace formula to attack the Principle of Functoriality. Many of the problems at this (early) stage concern the geometric terms in the trace formula. Most immediate perhaps is to extend Altug's application of Poisson summation from $\mathrm{GL}(2)$ to more general groups. One could then try to identify the contribution of the nontempered representations on the spectral side to the resulting Fourier transforms of geometric terms. An underlying problem here to formulate what might be the Beyond Endoscopic analogue of the Fundamental Lemma, an effort that will certainly require a command of Ngo's proof of the endoscopic fundamental lemma, and his related subsequent work. We shall discuss these problems, and possible directions that might lead to progress.