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量相关的代数非闭地面域上的代数簇上的有理点和有理曲线空间。在算术方面，人们对有理点的存在、它们在各种拓扑中的密度以及它们相对于高度的分布感兴趣。在几何方面，重点是双有理性质（例如有理性和有理连通性）、射影不变量（例如有效除数和充足除数的锥体）以及几何对应。在过去的几十年里，算术几何已成为最令人兴奋和快速发展的领域之一。在理解曲线算术方面已经取得了巨大进步。该提案的目标是增进我们对高维空间的理解。如果没有吸收其他数学分支的思想：超越理论、代数拓扑和调和分析，这些发展就不可能实现。作为回报，算术几何的进步对数学物理、动力系统、复杂分析产生了巨大影响。算术几何实验的需求导致了强大的计算工具和软件的开发，这些工具和软件现已广泛应用于密码学和数据分析等领域。