The Langlands program is a web of vast and far-reaching conjectures connecting seemingly distinct
areas of mathematics that are number theory and representation theory. At the heart of this program
lies an important principle called functoriality, that postulates the existence of deep relations between
the automorphic representations of different groups, as well as related central analytic objects called
automorphic L-functions. Recently, a new and particularly promising way to look at these notions,
and that has come to be called the relative Langlands program, has emerged. It essentially consists in
replacing groups by certain homogeneous spaces and to study their automorphic or local spectra.
As for the usual Langlands program, trace formulas are essential tools in the relative setting both
to tackle new conjectures than to deepen our understanding of the underlying principles. A main
theme of this proposal would be to make fundamental new contributions to the development of these
central objects in the local setting notably by: (1) Studying systematically the spectral expansions of
certain simple versions especially in the presence of an outer automorphism (twisted trace formula) (2)
Developing far-reaching local relative trace formulas for general spherical varieties making in particular
original new connections to the geometry of cotangent bundles. These progress would then be applied
to establish new and important instances of relative Langlands correspondences/functorialities. In
a slightly different but related direction, we also aim to study and develop other important tools of
harmonic analysis in a relative context, including Plancherel formulas and new kind of Paley-Wiener
theorems, with possible applications to new global comparison of trace formulas and factorization of
automorphic periods.
