  We introduce two classes of algebras coming from partial triangulations of
marked surfaces. The first one, called frozen algebra of a partial
triangulation, is generally of infinite rank and contains frozen Jacobian
algebras of triangulations of marked surfaces. The second one, called algebra
of a partial triangulation, is always of (explicit) finite rank and contains
classical Jacobian algebras of triangulations of marked surfaces and Brauer
graph algebras. We classify the partial triangulations, depending on the
complexity of their frozen algebras (some are free of finite rank, some are
lattices over a formal power series ring and most of them are not finitely
generated over their centre). For algebras of partial triangulations, we prove
that they are symmetric when the surface has no boundary. From a more
representation theoretical point of view, we prove that these algebras of
partial triangulations are of tame representation type and we define a
combinatorial operation on partial triangulation, generalizing Kauer moves of
Brauer graphs and flips of triangulations, which give derived equivalences of
the corresponding algebras.
