Arithmetic Analogue of Differential Algebraic Geometry

9730183 Buium在之前的研究中，Buium研究了一种类似于代数几何的方法，即用代数微分方程代替代数方程。这种几何可以称为微分代数几何，它的交换代数对应于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开发了一种微分代数几何的算术模拟。后者的应用包括“有效Manin-Mumford猜想”的证明和Manin核定理的算术模拟。这个项目的主要主题将是继续这一调查，特别是在算术方向上。这是在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25年里，它已经取得了革命性的成就。在它的起源中，它处理的是可以在平面上用最简单的方程定义的图形，即多项式。如今，该领域不仅使用代数的方法，而且使用分析和拓扑学的方法，反过来，在这些领域以及物理、理论计算机科学和机器人中也找到了应用。

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.