"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.

Broadening our scope to include the relative Langlands program (replacing groups by spherical varieties), in this series of talks we will revisit examples of relative trace formula comparisons that have appeared in the literature, and study the local "transfer operators" that realize these comparisons. Some structure will begin to emerge, that will be discussed further in subsequent talks, later in the semester.

The examples that will be discussed include:

(1) comparison between the Kuznetsov formula and the stable trace formula of $SL(2)$ (which first appeared in the thesis of Rudnick);

(2) comparison between the Kuznetsov formula and the relative trace formula for the variety $T\backslash PGL(2)$, where $T$ is a torus;

(3) comparison between the Kuznetsov formula for $GL(2)$ and the "trace formula" for a torus (Venkatesh's thesis).

Paradoxically (because functoriality was supposed to solve the problem of analytic continuation of L-functions), "beyond endoscopy" calls for the insertion of L-functions into trace formulas, and some treatment of their meromorphic continuation. This treatment is most successful when their functional equation can be expressed as a Poisson summation formula for certain "Hankel transforms" between spaces of orbital integrals. Thus, alongside the aforementioned transfer operators, we will also discuss two examples of Hankel transforms, namely:

(4) the Hankel transform for the standard L-function of $GL(n)$ on the Kuznetsov formula (contained in a paper of Jacquet);

(5) the Hankel transform for the symmetric square L-function of $GL(2)$ on the Kuznetsov formula (extracted from the Rankin–Selberg method).

Again, some structure will be visible, but we will also stumble on a precise local version of Sarnak's objection to the extension of these methods to higher symmetric powers.