A Celebration of Algebraic Geometry

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

The first decade of the twenty-first century has witnessed the solution of many fundamental and long-standing problems in algebraic geometry, including the proof of finite generation of the canonical ring by Birkar, Cascini, Hacon and McKernan and the discovery of a counterexample to Kodaira's conjecture by Voisin. Furthermore, the modern techniques developed earlier in the decade have led to far-reaching advances in the cohomology and the tautological rings of the moduli space of curves, Gromov-Witten theory, the study of rational curves and rational points on varieties and combinatorial descriptions of cohomological invariants of homogeneous varieties. The PIs are organizing a conference entitled "A celebration of algebraic geometry", on August 25-28, 2011 at Harvard University in Cambridge, Massachusetts, in order to reflect on these developments and to disseminate the advances to early career mathematicians. This grant will support young participants attending the conference. Algebraic geometry is the study of the solutions to a collection of polynomial equations. Even though algebraic geometry started more than two thousand years ago, with the study of the geometry of conic sections, such as circles, we still cannot completely answer many simple and fundamental questions. For example, given a collection of polynomials, with finitely many solutions, we would like to know the number of solutions. We know this number in many special cases, and these cases have interesting applications in engineering and biology. The first decade of the twenty-first century has seen tremendous advances on some of these fundamental problems. For example, Birkar, Cascini, Hacon and McKernan have shown that given a collection of polynomials, there is a nice way to present all of the solutions. This presentation allows us to answer many interesting questions. The PIs are organizing a conference to disseminate the advances of the last ten years to young mathematicians. This grant will support the participation of early career mathematicians at the conference.