Developing a Theory of Dynamical Complex Multiplication

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，即通过一些非平凡线性分数变换共轭的映射不仅保留了映射的动力学，而且保留了映射本身（例如，将其系数写成有理函数时）。由于大多数有理映射没有自同构，这可以被认为是复乘法的动态版本。该计划的智力价值在于，PI计划创建一个动态复杂乘法理论，其目标是开发足够的机器，以证明莫顿和西尔弗曼猜想对这类地图的证明。这项工作的更广泛影响来自教育和推广活动，重点是性别平等问题和支持妇女从事数学研究。建议的活动包括：(1)在夏威夷大学指导女性数学，从而扩大代表性不足群体的参与；(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系列讲座。

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.