This book is based on the Simons Symposia held in 2015. The proceedings in this volume focus on rational curves on higher-dimensional algebraic varieties and outlined applications of the theory of curves to arithmetic problems. There has been significant progress in this field, with major new results, which have given new impetus to the study of rational curves and spaces of rational curves on K3 surfaces and their higher-dimensional generalizations. One of the main recent insights covered here is that the geometry of rational curves is tightly coupled to properties of derived categories of sheaves on K3 surfaces. The implementation of this idea led to proofs of long-standing conjectures concerning birational properties of holomorphic symplectic varieties, which in turn should yield new theorems in arithmetic. These new insights are all covered in detail in this proceedings volume.

Fedor Bogomolov, Courant Institute of Mathematical Sciences, New York, NYBrendan Hassett, Brown University, Providence, Rhode IslandYuri Tshinkel,Courant Institute of Mathematical Sciences, New York, NY

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