Integral Points, Rational Curves and Entire Curves on Projective Varieties

射影簇上的积分点、有理曲线和整曲线

• 批准号: 1308737
• 负责人: Jason Starr
• 金额: $ 3.5万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-05-01 至 2014-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1308737&HistoricalAwards=false
• 关键词: Integral; Points; Rational; Curves; Entire

The month-long conference, "Rational Points, Rational Curves, and Entire Holomorphic Curves on Varieties", June 3-28, 2013, will be held at the Centre de Recherches Mathematiques in Montreal, Quebec, Canada. The URL for the conference website is as follows.http://www.crm.umontreal.ca/2013/Integral13/index_e.phpThe proposed activity is a summer school of 2-3 weeks comprised of a dozen mini-courses, followed by a week-long international workshop. There have been recent advances in seemingly separate fields: classification of complex projective varieties, existence and properties of solutions of Diophantine equations, and analytic behavior of entire curves in projective varieties. The moment is ripe for a summer school and workshop bringing together experts to synthesize these results, and train junior mathematicians in the new techniques. With the groundwork laid, we will emphasize open problems that seem amenable to solution: the holomorphic Lang conjecture about entire curves in general type varieties, potential density of rational points and rational curves on Calabi-Yau varieties, and the Grothendieck-Katz conjecture on algebraic solutions of differential equations.Systems of polynomial equations arise in all areas of mathematics, as well as areas of science and engineering as disparate as genomics and robot design. The oldest, and still most important, problem in this area is that of finding solutions to polynomial systems, particularly solutions that are fractions of whole numbers. Although surprising, this problem is closely connected to the related problem of interpolating any two given solutions by a system of solutions that are the outputs of a fraction of polynomials, or, more generally, an entire analytic function (the functions most amenable to study via "power series expansion"). There have been recent breakthroughs on both of these problems separately, as well as the interaction between them. This conference will feature training courses for young mathematicians, discussions with experts, and a public outreach lecture, as well as a planned volume to disseminate this work to an even broader audience.

为期一个月的会议“品种上的有理点、有理曲线和全全纯曲线”将于 2013 年 6 月 3 日至 28 日在加拿大魁北克省蒙特利尔的数学研究中心举行。 会议网站的URL如下：http://www.crm.umontreal.ca/2013/Integral13/index_e.php 拟议的活动是为期2-3周的暑期学校，由十几个迷你课程组成，随后是为期一周的国际研讨会。最近在看似独立的领域取得了进展：复杂射影簇的分类、丢番图方程解的存在性和性质，以及射影簇中整个曲线的分析行为。 现在时机已经成熟，可以举办暑期学校和研讨会，汇集专家来综合这些结果，并培训初级数学家掌握新技术。 奠定了基础后，我们将强调似乎可以解决的开放性问题：关于一般类型簇中整个曲线的全纯 Lang 猜想、Calabi-Yau 簇上有理点和有理曲线的势密度，以及代数上的 Grothendieck-Katz 猜想多项式方程组出现在数学的所有领域，以及基因组学和机器人设计等不同的科学和工程领域。 该领域最古老但仍然最重要的问题是寻找多项式系统的解，特别是整数分数的解。 尽管令人惊讶，但这个问题与通过解系统对任意两个给定解进行插值的相关问题密切相关，解系统是一部分多项式的输出，或者更一般地说，是整个解析函数（最适合通过“幂级数展开”）。 最近，这两个问题分别以及它们之间的相互作用都取得了突破。 这次会议将以针对年轻数学家的培训课程、与专家的讨论、公开讲座以及计划向更广泛的受众传播这项工作的计划为特色。