FRG: Collaborative Research: Algebraic Dynamics

代数动力学是研究数论、代数几何和离散动力系统界面上出现的问题的学科。簇的自映射迭代下的点的轨道对应于交换簇的有限生成的子群，在算术几何中关于簇上有理点、积分点和扭点的存在和分布的著名定理有自然的(大多是猜想的)代数动力学类似。研究者将利用数论、代数几何、丢番图逼近和模型理论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。离散动力学研究当一个函数重复应用于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，迭代以一种混乱的方式移动。代数动力学是将动力系统与代数和数论相结合的一个令人兴奋的新研究领域。当初始点为整数或有理数且函数由多项式给出时，研究人员将研究迭代轨道的数论性质。特别是，他们将研究(主要仍是猜测)数论中许多著名结果的动力学类似，这些结果描述了多项式方程组的整数解和有理解的分布。

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.