Integral Points, Rational Curves and Entire Curves on Projective Varieties

为期一个月的“有理点、有理曲线和全纯曲线上的变元”会议将于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。会议网站的网址是as follows.http://www.crm.umontreal.ca/2013/Integral13/index_e.phpThe提议的活动是一个为期2-3周的暑期学校，包括12个迷你课程，然后是一个为期一周的国际研讨会。最近在看似独立的领域中取得了一些进展：复射影簇的分类，丢番图方程解的存在性和性质，以及射影簇中整条曲线的解析行为。举办暑期班和研讨会的时机已经成熟，专家们将综合这些结果，并对初级数学家进行新技术方面的培训。随着基础的建立，我们将强调似乎可以解决的公开问题：关于一般类型变种中的整条曲线的全纯Lang猜想，关于Calabi-Yau变种上的有理点和有理曲线的势密度，以及关于微分方程组的代数解的Grothendieck-Katz猜想。多项式方程组出现在数学的所有领域，以及从基因组学到机器人设计的不同的科学和工程领域。这个领域中最古老也是最重要的问题是寻找多项式系统的解，特别是整数的分数解。尽管令人惊讶，但这个问题与相关问题密切相关，即通过一个解系对任意两个给定解进行内插，该解系是一个多项式的分式的输出，或者更一般地，是整个解析函数(最易于研究的函数，通过“幂级数展开”)。最近在这两个问题上以及它们之间的相互作用上都有了突破。这次会议将包括青年数学家的培训课程、与专家的讨论和公众宣传讲座，以及计划向更广泛的受众传播这项工作的卷。

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.