Points on Curves Over Finite Fields and Motivic Stabilization

研究了固定有限域上曲线上的点的分布以及Grothendieck环上的模空间序列，特别是曲线序列的稳定性。固定有限域上一族曲线上的点的分布问题是相当神秘的，即使在启发式水平上也是如此，但许多最近的发展允许有趣地访问特殊情况。这些案例可以为一般理论的发展提供基础。最近，有限域上簇的计点的相关问题被确定，以反映簇的Grothendieck环的更深层次的结构，这将在本项目中得到研究和进一步发展。关于一个方程的一个基本问题是：它有多少解？许多方程自然而然地归入相似的方程家族，例如，二次方程中所有项的次数至多为两次。这个项目研究如何回答“有多少解决方案？”家庭内部的变化以及随着家庭变得更加复杂而形成的一致的模式。该项目还包括指导本科生，支持建立网络和辅导女性数学的基础设施，向中学生、高中生、大学非数学专业的学生和普通公众传播数学思想，以及指导和培训研究生

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