Points on Curves Over Finite Fields and Motivic Stabilization

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The project is to study the distribution of points on curves over a fixed finite field and stabilization of sequences of moduli spaces, especially of curves, in the Grothendieck ring of varieties. The question of distribution of points on curves in a family over a fixed finite field is quite mysterious, even on the heuristic level, but many recent developments allow interesting access to special cases. These cases can provide the groundwork for development of the general theory. Related questions of counting points on varieties over finite fields have recently been determined to reflect a deeper structure in the Grothendieck ring of varieties, which will be studied and further developed in this project.A basic question about an equation is: how many solutions does it have? Many equations naturally fall into families of similar equations, for example, quadratic equations are ones in which all terms have degree at most two. This project studies how answers to the question of "how many solutions?" vary within a family and what consistent patterns develop as the family gets more complicated. The project also involves mentoring undergraduates, supporting infrastructure for networking and mentoring of women in mathematics, dissemination of mathematical ideas to middle school students, high school students, college non-math majors, and the general public, and mentoring and training graduate students