Surveys on Recent Developments in Algebraic Geometry

Description:

The algebraic geometry community has a tradition of running a summer research institute every ten years. During these influential meetings a large number of mathematicians from around the world convene to overview the developments of the past decade and to outline the most fundamental and far-reaching problems for the next. The meeting is preceded by a Bootcamp aimed at graduate students and young researchers. This volume collects ten surveys that grew out of the Bootcamp, held July 6–10, 2015, at University of Utah, Salt Lake City, Utah. These papers give succinct and thorough introductions to some of the most important and exciting developments in algebraic geometry in the last decade. Included are descriptions of the striking advances in the Minimal Model Program, moduli spaces, derived categories, Bridgeland stability, motivic homotopy theory, methods in characteristic and Hodge theory. Surveys contain many examples, exercises and open problems, which will make this volume an invaluable and enduring resource for researchers looking for new directions.

subjects: Geometry, algebraic,  Algebraic Geometry,  Congresses,  Algebraic geometry -- Curves -- Families, moduli (algebraic),  Algebraic geometry -- Birational geometry -- Minimal model program (Mori theory, extremal rays),  Algebraic geometry -- Birational geometry -- Rationality questions,  Algebraic geometry -- Families, fibrations -- Variation of Hodge structures,  Algebraic geometry -- Projective and enumerative geometry -- Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants,  Algebraic geometry -- Surfaces and higher-dimensional varieties -- Vector bundles on surfaces and higher-dimensional varieties, and their moduli,  Algebraic geometry -- Arithmetic problems. Diophantine geometry -- Positive characteristic ground fields,  Commutative algebra -- Homological methods -- Syzygies, resolutions, complexes,  $K$-theory -- Higher algebraic $K$-theory -- $Q$- and plus-constructions,  Dynamical systems and ergodic theory -- Dynamical systems with hyperbolic behavior -- Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.).,  Curves,  Families, moduli (algebraic),  Birational geometry,  Minimal model program (Mori theory, extremal rays),  Rationality questions,  Families, fibrations,  Variation of Hodge structures,  Projective and enumerative geometry,  Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants,  Surfaces and higher-dimensional varieties,  Vector bundles on surfaces and higher-dimensional varieties, and their moduli,  Arithmetic problems. Diophantine geometry,  Positive characteristic ground fields,  Commutative algebra,  Homological methods,  Syzygies, resolutions, complexes,  $K$-theory,  Higher algebraic $K$-theory,  $Q$- and plus-constructions,  Dynamical systems and ergodic theory,  Dynamical systems with hyperbolic behavior,  Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)