I will provide a general introduction to tensor models and their applications to quantum gravity. Initially developed in the context of random geometry, as a generalization of the matrix models approach to 2d quantum gravity, tensor models and their large N expansion have more recently been taken advantage of in the context of holography. In particular, the "near AdS_2 / near CFT_1 correspondence" establishes a connection between strongly-coupled and explicitly solvable large N quantum mechanics and Jackiw-Teitelboim gravity in d=2.

The conformal bootstrap aims to systematically constrain CFTs based on crossing symmetry and
unitarity. In this talk I will introduce a new approach to extract information from the crossing symmetry
sum rules, based on the construction of linear functionals with certain positivity properties. I show
these functionals allow us to derive a class of optimal bounds on CFT data, and also act as an ideal basis
for obtaining other bounds numerically. Furthemore I will argue that special extremal solutions to