FRG: Collaborative Research: Algebraic Dynamics

代数动力学是研究发生在数论、代数几何和离散动力系统界面上的问题。在变种的自映射的迭代下点的轨道对应于有限生成的阿贝尔变种的子群，并且在算术几何中关于变种上的有理点、积分点和扭转点的存在和分布的著名定理有自然的（主要是推测的）代数动态类似物。研究者将使用从数论、代数几何、丢番图近似和模型理论中提取的工具来研究这些代数动力学问题。他们还将研究相关的模问题，并将研究动态模空间和动态模曲线的几何和算术性质。离散动力学研究了当一个函数被重复地应用于一个初始点时会发生什么。对于某些点，行为表现良好，而对于其他点，迭代以混乱的方式移动。代数动力学是一个令人兴奋的新研究领域，它将动力系统与代数和数论相结合。研究者将研究当初始点为整数或有理数，函数为多项式给出时迭代轨道的数论性质。特别是，他们将研究数论中许多著名结果的动态类似物（主要是推测性的），这些结果描述了多项式方程系统的整数和有理解的分布。

Algebraic dynamics is the study of problems that occur on the interface of number theory, algebraic geometry, and discrete dynamical systems. Orbits of points under iteration of a self-map of a variety correspond to finitely generated subgroups of abelian varieties, and there are natural (mostly conjectural) algebraic dynamical analogues of famous theorems in arithmetic geometry regarding the existence and distribution of rational, integral, and torsion points on varieties.The investigators will study these algbebraic dynamical questions using tools drawn from number theory, algebraic geometry, Diophantine approximation, and model theory. They will also study associated moduli problems and will investigate geometric and arithmetic properties of dynamical moduli spaces and dynamical modular curves.Discrete dynamics studies what happens when a function is repeatedly applied to an initial point. For some points, the behavior is well-behaved, while for other points the iterates move around in a chaotic fashion. Algebraic dynamics is an exciting new area of research that amalgamates dynamical systems with algebra and number theory. The investigators will study number theoretic properties of the orbit of iterates when the initial point is an integer or a rational number and the function is given by polynomials. In particular, they will study (mostly still conjectural) dynamical analogues of many famous results in number theory that describe the distribution of integer and rational solutions to systems of polynomial equations.