A1-Homotopy Theory and Applications to Enumerative Geometry and Number Theory

该奖项支持一项研究计划，该计划涉及一种丰富的计数形式，以研究方程的解及其形成的空间。如果一组方程的解可以用通常的计数数来表示，或者如果需要实数，或者如果一个人必须使用虚数，这是很重要的。丰富的计数检测到了这样的差异。在某些情况下，它与方程实解空间形状中d维的孔洞的数量密切相关。这个项目利用了丰富的计数的力量，揭示了在数论和代数几何中的潜在应用。该奖项还将支持建立一支强大和多样化的数学劳动力队伍。这将包括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的持续计划。在这个项目中，PI将促进与高中学生和教师的合作项目，必要时提供背景材料。暑期项目的毕业生将被鼓励继续为本科生提供研究体验，这将提供进一步的数学培训和研究指导。本研究在Morel和Voevodsky的A1-同伦理论框架下，利用上同调理论和同伦方法研究数论和代数几何问题。该项目使用稳定的A1-同伦理论在非代数闭域和整数环上产生计数几何的结果。定义在一般场上的新的Gromov-Witten不变量有可能满足穿墙公式、外科公式和WDVV方程。为此，该项目研究了一般场上的自旋概念。Weil猜想将有限域上方程的解的个数与其复点的拓扑联系起来：有限域上的变种的Zeta函数同时是其定义方程的解的个数的母函数和上同调群的自同态的特征多项式的乘积。这些上同调群的秩是相关复流形的Betti数。Zeta函数的对数导数被丰富为具有Grothendieck-Witt群中的系数的幂级数，从而产生与相关实流形的联系。该项目旨在加强我们对Zeta函数及其应用的对数导数的控制。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This award supports a research program involving an enriched form of counting to study the solutions of equations and the spaces they form. It matters if the solution to a set of equations can be expressed using the usual counting numbers, or if real numbers are required, or if one must use imaginary numbers. The enriched count detects such differences. In some cases, it is closely connected to the number of holes of dimension d in the shape of a space of real solutions to the equations. This project exploits the power of the enriched count, exposing potential applications in number theory and algebraic geometry. The award will also support a pipeline for a strong and diverse mathematical workforce. This will involve a continuing program of week-long summer math jobs for gifted high school students from diverse backgrounds. During this program, the PI will facilitate collaborative projects with high school student and teachers, providing background material as necessary. Graduates from the summer program will be encouraged to continue on to a Research Experience for Undergraduates that will provide further mathematical training and research mentorship. The proposed research studies number-theoretic and algebro-geometric questions using cohomology theories and homotopical methods in the framework of Morel and Voevodsky's A1-homotopy theory. The project uses stable A1-homotopy theory to produce results in enumerative geometry over non-algebraically closed fields and rings of integers. New Gromov--Witten invariants defined over general fields have the potential to satisfy wall-crossing formulas, surgery formulas, and WDVV equations. For this, the project studies notions of spin over general fields. The Weil conjectures connect the number of solutions to equations over finite fields to the topology of their complex points: The zeta function of a variety over a finite field is simultaneously a generating function for the number of solutions to its defining equations and a product of characteristic polynomials of endomorphisms of cohomology groups. The ranks of these cohomology groups are the Betti numbers of the associated complex manifold. The logarithmic derivative of the zeta function is enriched to a power series with coefficients in the Grothendieck--Witt group, producing a connection with the associated real manifold. This project aims to increase our control over this logarithmic derivative of the zeta function and its 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.