Perspectives on Complex Algebraic Geometry

复杂代数几何的观点

• 批准号: 1502166
• 负责人: Aise de Jong
• 金额: $ 2.45万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-02-15 至 2016-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1502166&HistoricalAwards=false
• 关键词: Perspectives; Complex; Algebraic; Geometry

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/.

该奖项为研讨会提供资金“复杂代数几何的观点”将于2015年5月22日至25日在哥伦比亚大学，纽约，纽约举行。代数几何是研究多元多项式方程的解的学科。 由于多项式出现在每一个可以用数值研究的主题中，代数几何在数学中起着核心作用，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，在过去的一百年里取得了一些最重要的数学成就，并且今天仍然是一个蓬勃发展的重要领域。在代数几何中，复代数几何扮演着特殊的角色。也就是说，解决方案集的复杂结构允许广泛的技术，包括几何，分析和拓扑结构。一些最强大的技术涉及霍奇理论的使用，该理论使用紧致流形上的调和理论来关联解集的拓扑、复杂结构和代数性质。近年来，与物理学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都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发展的技术解决了。 本次研讨会的目的是回顾复代数几何领域的最新发展，重点讨论以下三个主题：（1）代数曲面和4-流形的拓扑和几何，（2）向量丛和G-丛，（3）Hodge理论的几何应用。这些都是中心议题在该领域的复杂代数几何，已经看到了一些有趣的最近的发展，并躺在交叉的一些领域的数学，但已不太好地代表最近的讲习班。研讨会将汇集该领域的一些主要专家以及一些年轻的研究人员，他们将进一步推动这些方向的发展。有关会议的更多详细信息，请访问其网站：https://sites.google.com/site/complexalgebraicgeometry/。