Spaces of rational curves and diophantine geometry

有理曲线空间和丢番图几何

• 批准号: 1160859
• 负责人: Yuri Tschinkel
• 金额: $ 20万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-08-01 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1160859&HistoricalAwards=false
• 关键词: Spaces; rational; curves; diophantine; geometry

The main questions of this proposal concern rational points and spaces of rational curves on algebraic varieties over algebraically nonclosed ground fields in relation to geometric or topological invariants. On the arithmetic side, one is interested in existence of rational points,their density in various topologies, and their distribution with respect to heights. On the geometric side,the focus is on birational properties, such as rationality and rational connectedness, on projective invariants, such as the cones of effective and ample divisors, and on geometric correspondences. In the last decades, arithmetic geometry has become one of the most exciting and rapidly growing fields. There has been tremendous progress in understanding the arithmetic of curves. The goal of this proposal is to advance our understanding of higher-dimensional spaces. These developments would not be possible without the assimilation of ideas from other branches of mathematics: transcendence theory, algebraic topology, and harmonic analysis. In return, advances in arithmetic geometry have had strong impact in mathematical physics, dynamical systems, complex analysis. The need for experimentation in arithmetic geometry has lead to the development of powerful computational tools and software, which are now widely used, e.g., in cryptography and data analysis.

该提案的主要问题涉及与几何或拓扑不变的代数未公开的地面场上代数曲线上有理曲线的合理点和空间。在算术方面，人们对存在理性点，它们在各种拓扑中的密度以及它们在高度方面的分布感兴趣。在几何方面，重点是对射影不变性的birational属性，例如理性和理性联系，例如有效和足够的除数的锥，以及几何对应关系。在过去的几十年中，算术几何形状已成为最令人兴奋和快速发展的领域之一。理解曲线的算术方面取得了巨大进展。该提案的目的是提高我们对高维空间的理解。如果没有其他数学分支的思想：超越理论，代数拓扑和谐波分析，这些发展是不可能的。作为回报，算术几何形状的进步对数学物理学，动力学系统，复杂分析产生了强烈的影响。在算术几何形状中进行实验的需求导致了强大的计算工具和软件的开发，例如，在加密和数据分析中广泛使用。