Developing a Theory of Dynamical Complex Multiplication

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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.