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，“合理观点，理性曲线和整个品种的全体形态曲线”将于2013年6月3日至28日在加拿大魁北克蒙特利尔的De Rechherches Mathematiques举行。 会议网站的URL如下。在看似独立的领域中，最近的进步已经取得了进步：复杂的投射品种的分类，Diophantine方程解的存在和特性的分类，以及整个曲线曲线的分析行为。 这个时刻已经成熟，暑期学校和研讨会将专家汇集在一起​​，以综合这些结果，并培训初级数学家的新技术。 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作为科学和工程领域，就像基因组学和机器人设计一样。 在该领域中，最古老，最重要的问题是找到对多项式系统的解决方案，尤其是解决方案的解决方案。 尽管令人惊讶，但这个问题与通过解决多项式的输出的解决方案系统插值任何两个给定解决方案的相关问题密切相关，或者更普遍地是整个分析功能（通过“功率系列扩展”研究的功能最适合研究）。 最近在这两个问题上都出现了突破，以及它们之间的相互作用。 这次会议将为年轻的数学家提供培训课程，与专家的讨论以及公开展开演讲，以及计划的卷，以向更广泛的受众传播这项工作。