FRG: Collaborative Research: Algebraic Dynamics

FRG：合作研究：代数动力学

• 批准号: 0854746
• 负责人: Lucien Szpiro
• 金额: $ 19.21万
• 依托单位: CUNY Graduate School University Center
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-09-01 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0854746&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Algebraic; Dynamics

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.

代数动力学是对数字理论，代数几何和离散动态系统界面上出现问题的研究。 在迭代中的自图下的轨道轨道对应于有限生成的亚伯利亚品种的亚组，并且在算术几何学上有自然的（主要是猜想的）代数的代数动态类似物类似于算术的存在和分布在有理由，整体上的分布，并研究了这些algbebbebbebbebbebbebe。代数几何形状，二氧化苯胺近似和模型理论。他们还将研究相关的模量问题，并将研究动态模量空间的几何和算术特性和动态模块化曲线。DiscreteDynamics研究，当函数反复应用于初始点时会发生什么。在某些方面，行为表现良好，而对于其他要点，迭代的方式是混乱的。代数动力学是一个令人兴奋的研究领域，该领域与代数理论和数字理论合并动态系统。研究人员将研究迭代轨迹的数量理论特性，而初始点是整数或有理数的，并且该函数由多项式给出。特别是，他们将研究许多著名结果理论中许多著名结果的动态类似物，这些动态类似物描述了整数和理性解决方案对多项式方程系统的分布。