  We lay the groundwork for a UV-complete formulation of the Euclidean
Jackiw-Teitelboim two-dimensional models of quantum gravity when the boundary
lengths are finite, emphasizing the discretized approach. The picture that
emerges is qualitatively new. For the disk topology, the problem reduces to
counting so-called self-overlapping curves, that are closed loops that bound a
distorted disk, with an appropriate multiplicity. We build a matrix model that
does the correct counting. The theories in negative, zero and positive
curvatures have the same UV description but drastically different macroscopic
properties. The Schwarzian theory emerges in the limit of very large and
negative cosmological constant in the negative curvature model, as an effective
theory valid on distance scales much larger than the curvature length scale. In
positive curvature, we argue that large geometries are ubiquitous and that the
theory exists only for positive cosmological constant. Our discussion is
pedagogical and includes a review of several relevant topics.
