Hyperbolicity of moduli spaces: complex differential geometry meets
In several branches of geometry, a space is called a moduli space if it parametrizes some class of geometric objects. Examples of such objects are manifolds, varieties, maps, etc. A space is usually called hyperbolic, if it is similar in some sense to a Riemann surface, or equivalently to a smooth projective algebraic curve, of genus at least two.
I will give a brief introduction to hyperbolicity properties of moduli spaces, a classical but still intensely investigated area. Although, I plan to mention the recent research directions in the end, I will focus on the basic ideas of the field. I intend to emphasize that many of the statements can be approached either from a holomorphic, or an algebraic, or a differential geometric point of view. For most of the talk, I will only assume an introductory knowledge of manifolds and holomorphic functions.