Points on Curves Over Finite Fields and Motivic Stabilization

该项目是研究在一个固定的有限域上的曲线上的点的分布和稳定的序列的模空间，特别是曲线，在Grothendieck环的品种。 在一个固定的有限域上，一个族中的曲线上的点的分布问题是相当神秘的，即使在启发式的水平上也是如此，但是最近的许多发展允许有趣的特殊情况。 这些案例可以为一般理论的发展提供基础。 有限域上簇上的点的计数问题反映了簇的Grothendieck环的一个更深层次的结构，本项目将对此进行进一步的研究和发展。方程的一个基本问题是：它有多少解？ 许多方程自然地落入类似方程的族中，例如，二次方程是其中所有项的次数最多为2的方程。 本项目研究如何回答“有多少个解决方案？“在一个家庭中各不相同，随着家庭变得更加复杂， 该项目还涉及指导本科生，支持网络基础设施和指导数学领域的妇女，向中学生、高中生、大学非数学专业学生和公众传播数学思想，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