Perspectives on Complex Algebraic Geometry

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This award provides funding for the workshop "Perspectives on Complex Algebraic Geometry" to take place May 22-25, 2015 at Columbia University, New York, New York. Algebraic Geometry is the study of the solutions of polynomial equations in several variables. As polynomials arise in every topic that can be studied numerically, algebraic geometry plays a central role within mathematics and has close ties to the sciences. The field traces its roots back to the foundations of mathematics, has led to some of the most significant mathematical accomplishments in the past hundred years, and continues to be a burgeoning and vital field today. Within the field of algebraic geometry, complex algebraic geometry plays a special role. Namely, the complex structure of the solution sets allows for a wide range of techniques, including those from geometry, analysis, and topology. Some of the most powerful techniques involve the use of Hodge theory, which uses harmonic theory on compact manifolds to relate the topology, complex structure, and algebraic properties of the solutions sets. Recently, the connections with physics have played an increasingly important role in algebraic geometry. Many questions in physics can be phrased naturally in the language of differential geometry. Through deep and surprising connections that exist between differential geometry and complex algebraic geometry, some of these questions can be addressed using the techniques of algebraic geometry. Questions posed by physicists have been solved using techniques developed by algebraic geometers. In turn, recent developments in physics have led to astonishing new results and open problems in algebraic geometry.The purpose of the workshop is to survey the recent developments in the field of complex algebraic geometry with a focus on the following 3 main topics: (1) Topology and geometry of algebraic surfaces and 4-manifolds, (2) Vector bundles and G-bundles, and (3) Geometric applications of Hodge theory. These are central topics in the field of complex algebraic geometry, that have seen a number of interesting recent developments, and lie at the intersection of a number of fields of mathematics, but which have been less well represented lately in terms of workshops. The workshop will bring together some of the leading experts in the field, as well as a number of young researchers, who will further propel developments in these directions. More details on the conference can be found at its website: https://sites.google.com/site/complexalgebraicgeometry/.