  We present a variety of geometrical and combinatorial tools that are used in
the study of geometric structures on surfaces: volume, contact, symplectic,
complex and almost complex structures. We start with a series of local rigidity
results for such structures. Higher-dimensional analogues are also discussed.
Some constructions with Riemann surfaces lead, by analogy, to notions that hold
for arbitrary fields, and not only the field of complex numbers. The Riemann
sphere is also defined using surjective homomorphisms of real algebras from the
ring of real univariate polynomials to (arbitrary) fields, in which the field
with one element is interpreted as the point at infinity of the Gaussian plane
of complex numbers. Several models of the hyperbolic plane and hyperbolic
3-space appear, defined in terms of complex structures on surfaces, and in
particular also a rather elementary construction of the hyperbolic plane
usingreal monic univariate polynomials of degree two without real roots.
Several notions and problems connected with conformal structures in dimension 2
are discussed, including dessins d'enfants, the combinatorial characterization
of polynomials and rational maps of the sphere, the type problem,
uniformization, quasiconformal mappings, Thurston's characterization of Speiser
graphs, stratifications of spaces of monic polynomials, and others. Classical
methods and new techniques complement each other. The final version of this
paper will appear as a chapter in the Volume Surveys in Geometry. II (ed. A.
Papadopoulos), Springer Nature Switzerland, 2024.
