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; 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.