Arithmetic Analogue of Differential Algebraic Geometry

微分代数几何的算术模拟

• 批准号: 9730183
• 负责人: Alexandru Buium
• 金额: $ 10.53万
• 依托单位: University of New Mexico
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-06-01 至 1998-12-07
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9730183&HistoricalAwards=false
• 关键词: Arithmetic; Analogue; Differential; Algebraic; Geometry

9730183 Buium In previous research, Buium investigated an analogue of algebraic geometry in which algebraic equations are replaced by algebraic differential equations. This geometry may be called differential algebraic geometry, and its commutative algebra counterpart is the Ritt-Kolchin differential algebra. Applications of Buium's work include the ``effective geometric Lang conjecture.'' In subsequent research, Buium developed an arithmetic analogue of differential algebraic geometry. Applications of the latter included a proof of the "effective Manin-Mumford conjecture" and an arithmetic analogue of Manin's theorem of the Kernel. The main theme of this project will be continuing this line of investigation, especially in the arithmetic direction. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.

小行星9730183 在以前的研究中，Buium研究了代数几何的一个模拟，其中代数方程被代数微分方程取代。这种几何可以称为微分代数几何，它的交换代数对应物是Ritt-Kolchin微分代数。Buium工作的应用包括“有效几何Lang猜想”。在随后的研究中，Buium开发了微分代数几何的算术模拟。后者的应用包括“有效的马宁-芒福德猜想”的证明和马宁核定理的算术模拟。这个项目的主题将继续这条调查路线，特别是在算术方向。 这是代数几何领域的研究。代数几何是现代数学中最古老的部分之一，但在过去的四分之一个世纪里，它已经有了革命性的发展。在其起源，它处理的数字，可以定义在平面上的最简单的方程，即多项式。如今，该领域不仅使用代数方法，还使用分析和拓扑学方法，相反，这些方法在这些领域以及物理学，理论计算机科学和机器人学中也得到了应用。