Perspectives on Complex Algebraic Geometry

该奖项为将于2015年5月22日至25日在纽约哥伦比亚大学举行的“复杂代数几何透视”研讨会提供资金。代数几何是研究多变量多项式方程解的学科。随着多项式出现在每一个可以用数值方法研究的课题中，代数几何在数学中扮演着中心角色，并且与科学有着密切的联系。该领域的根源可以追溯到数学的基础，在过去的一百年里产生了一些最重要的数学成就，并且今天仍然是一个蓬勃发展和充满活力的领域。在代数几何领域中，复代数几何起着特殊的作用。也就是说，解决方案集的复杂结构支持广泛的技术，包括来自几何、分析和拓扑的技术。一些最强大的技术涉及到Hodge理论的使用，它使用紧流形上的调和理论来联系解集的拓扑结构、复杂结构和代数性质。近年来，与物理的联系在代数几何中起着越来越重要的作用。物理学中的许多问题可以用微分几何的语言自然地表述出来。通过微分几何和复杂代数几何之间存在的深刻而令人惊讶的联系，其中一些问题可以使用代数几何的技术来解决。物理学家提出的问题已经用代数几何学家开发的技术解决了。反过来，物理学的最新发展导致了代数几何中惊人的新结果和开放的问题。研讨会的目的是回顾复杂代数几何领域的最新发展，重点关注以下3个主要主题：(1)代数曲面和4流形的拓扑和几何；(2)向量束和g束；(3)Hodge理论的几何应用。这些都是复杂代数几何领域的核心主题，最近有了一些有趣的发展，并且处于许多数学领域的交叉点，但最近在研讨会方面却没有得到很好的代表。该研讨会将汇集该领域的一些主要专家以及一些年轻研究人员，他们将进一步推动这些方向的发展。有关会议的更多细节，请访问其网站：https://sites.google.com/site/complexalgebraicgeometry/。

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/.