Motivic Homotopy Theory and Applications to Enumerative Geometry

这个项目研究某些方程的解的数量，使用与这些方程相关的空间形状，以及这个形状本身。这类解的个数有一个新的不变性。这种不变性不仅适用于复数的解的个数，也适用于普通分数的解，比如1 / 2。利用Morel和Voevodsky的a1 -同伦理论得到了它。此外，还为来自不同背景的天才高中生提供为期一周的暑期数学工作，以支持数学教育。在项目期间的每个夏天，大约八名高中生将研究一个重要的数学问题，学习必要的背景材料，并作为一个小组解决它。他们将由两名高中老师陪同。为本科生提供研究经验，旨在为该计划的毕业生提供继续数学训练和研究指导。a1 -同伦理论是由Morel和Voevodksy在20世纪90年代末提出的，它允许将代数拓扑的工具成功地引入到多项式方程解的研究中。PI及其合作者正在研究a1 -同伦理论与枚举几何中的经典问题之间的相互作用，例如“在空间中有多少条线与四条线相交？”a1 -同伦理论在非常一般的基格式上，特别是在任何域上都能很好地发挥作用，从而在双线性形式的非代数闭域上得到枚举结果。本项目寻找与特征类、Gromov—Witten理论和zeta函数相关的结果，并开发由这些应用提出的动机同伦理论中的工具。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project studies the number of solutions to certain equations, using the shape of spaces associated to these equations, as well as this shape itself. There is a new invariance property of the number of such solutions. This invariance not only applies to the number of solutions in the complex numbers, but also solutions which are ordinary fractions, such as one half. It is obtained by using A1-homotopy theory due to Morel and Voevodsky. Mathematics education is furthermore supported by continuing a program of week-long summer math jobs for gifted high school students from diverse backgrounds. During each of the summers of the project period, approximately eight high school students will work on an important mathematical problem, learning the background material as necessary, and solving it as a group. They will be accompanied by two high school teachers. A Research Experience for Undergraduates aimed at the graduates of the program is provided to continue mathematical training and provide research mentorship. A1-homotopy theory was introduced by Morel and Voevodksy in the late 1990's and allows the successful import of tools from algebraic topology into the study of solutions to polynomial equations. The PI and collaborators are studying the interaction between A1-homotopy theory and classical questions from enumerative geometry such as "How many lines meet four lines in space?" A1-homotopy theory functions well over very general base schemes and in particular over any field, resulting in enumerative results over non-algebraically closed fields valued in bilinear forms. This project searches for such results connected with characteristic classes, Gromov--Witten theory, and zeta functions and develops tools in motivic homotopy theory suggested by these applications.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.