Dynamical Teichmüller spaces for rational and transcendental maps

Every smooth orientable surface can be endowed with a complex structure, turning it into a Riemann surface. However, this complex structure is usually not unique, and beginning with Riemann in the 1830s, mathematicians have been studying the space of conformal structures of a surface, called its moduli space. It turns out that moduli space is a quotient of Teichmüller space, which has a very nice structure: It is a complex manifold and carries a natural metric. Since its introduction in the early 20th century, Teichmüller spaces have been very successful in studying the geometry and deformation of surfaces, and versions of it have been introduced as tools to study deformations in other contexts such as Kleinian groups and complex dynamics.
In this talk, I will review some basics of the classical theory and explain the construction, properties, and applications of dynamical moduli and Teichmüller spaces for rational maps, introduced by McMullen and Sullivan in the 1980s and 1990s. Lastly, if time allows, I will talk about some recent work related to dynamical Teichmüller spaces for entire transcendental maps.