Jayce R. Getz (Duke University)

Triple product L-functions are the Langlands L-functions attached to a
triple of automorphic representations of GL_{r_i} for positive integers
r_1,r_2,r_3 and the tensor product representation

GL_{r_1} \times GL_{r_2} \times GL_{r_3} \lto \GL_{r_1r_2r_3}.

If we knew their analytic properties then by the converse theorem one
would be able to deduce the automorphy of Rankin-Selberg products of
automorphic representations.  This corresponds to the special case
r_3=1.   This in turn would make huge inroads into Langlands
functoriality.  I will report on joint work with Pam Gu, Chun-Hsien Hsu,
and Spencer Leslie in which we construct an integral representation
related to triple product L-functions, generalizing work of Garrett,
Piatetski-Shapiro, and Rallis in the case r_1=r_2=r_3=2.  I will also
comment on the obstacles to using this integral representation to obtain
the functional equation of triple product L-functions, and how one might
overcome them