Developing a Theory of Dynamical Complex Multiplication

发展动态复数乘法理论

• 批准号: 1102858
• 负责人: Michelle Manes
• 金额: $ 9.98万
• 依托单位: University of Hawaii
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-08-15 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1102858&HistoricalAwards=false
• 关键词: Developing; Theory; Dynamical; Complex; Multiplication

In her thesis, the PI pioneered the study of the dynamics of rational maps having a nontrivial automorphism, that is maps for which conjugating by some nontrivial linear fractional transformation preserves not just the dynamics of the map but the map itself (its coefficients when written as a rational function, say). Since most rational maps have no automorphisms, this might be considered a dynamical version of complex multiplication. The intellectual merit of the proposed project lies in the PI's plan to create a theory of dynamical complex multiplication, with the goal of developing enough machinery that a proof of a Conjecture of Morton and Silverman for this family of maps is within reach. Broader impacts of the work emerge from education and outreach activities, with a focus on gender equity issues and supporting women in research mathematics. Proposed activities include:(1) mentoring women in mathematics at the University of Hawaii and thus broadening participation of underrepresented groups;(2) establishing the Math Teachers' Circle of Hawaii; and (3) continuing the University of Hawaii Mathematics Department's Distinguished Lecture Series.

在她的论文中，PI率先研究了具有非平凡自同构的有理映射的动力学，即通过某种非平凡的线性分式变换进行共轭的映射不仅保持了映射的动力学，而且保持了映射本身(例如，其系数被写为有理函数时的系数)。由于大多数有理映射没有自同构，这可以被认为是复数乘法的动态版本。提议的项目的智力价值在于PI计划创建动态复数乘法理论，目标是开发足够的机器，以便证明Morton和Silverman关于这一映射族的猜想是触手可及的。这项工作的更广泛影响来自教育和外联活动，重点是性别平等问题和支持妇女进行数学研究。拟议的活动包括：(1)在夏威夷大学指导妇女数学，从而扩大代表不足群体的参与；(2)建立夏威夷数学教师圈；(3)继续夏威夷大学数学系的杰出讲座系列。