**Jaikrishnan Janardhanan**

Indian Institute of Technology, Madras

May 25, 2017

** Finiteness Theorems for Holomorphic Mappings from Products**:
It is somewhat surprising that the space of non-constant/dominant holomorphic mappings between complex manifolds is often empty or at most finite. The classical theorems of Liouville and Picard are examples of this phenomenon. In the context of compact Riemann
surfaces, the famous de Franchis theorem asserts that the set of
non-constant holomorphic mappings between two compact Riemann surfaces of genus higher than 2 is a finite set. In this talk, we outline some results of this type for (not-necessarily compact) Riemann surfaces and products of Riemann surfaces. We will also highlight the role that hyperbolicity (in the sense of Kobayashi) plays in such results.