Foliation Theory in Algebraic Geometry Featuring a blend of original research papers and comprehensive surveys from an international team of leading researchers in the thriving fields of foliation theory holomorphic foliations and birational geometry this book presents the proceedings of the conference Foliation Theory in Algebraic Geometry hosted by the Simons Foundation in New York City in September 2013. Topics covered include: Fano and del Pezzo foliations; the cone theorem and rank one foliations; the structure of symmetric differentials on a smooth complex surface and a local structure theorem for closed symmetric differentials of rank two; an overview of lifting symmetric differentials from varieties with canonical singularities and the applications to the classification of AT bundles on singular varieties; an overview of the powerful theory of the variety of minimal rational tangents introduced by Hwang and Mok; recent examples of varieties which are hyperbolic and yet the Green Griffiths locus is the whole of X; and a classification of psuedoeffective codimension one distributions. Foliations play a fundamental role in algebraic geometry for example in the proof of abundance for threefolds and to a solution of the Green Griffiths conjecture for surfaces of general type with positive Segre class. The purpose of this volume is to foster communication and enable interactions between experts who work on holomorphic foliations and birational geomet ry and to bring together leading researchers to demonstrate the powerful connection of ideas methods and goals shared by these two areas of study. On Fano Foliations 2 Rational Curves on Foliated Varieties Local Structure of Closed Symmetric 2 Differentials Aspects of the Geometry of Varieties with Canonical Singularities Geometric Structures and Substructures on Uniruled Projective Manifolds Foliations Shimura Varieties and the Green Griffiths Lang Conjecture On the Structure of Codimension I Foliations with Pseudoeffective Conormal Bundle.