CAREER: Etale and Motivic Homotopy Theory and Applications to Arithmetic Geometry

职业：基元同伦理论及其在算术几何中的应用

• 批准号: 2001890
• 负责人: Kirsten Wickelgren
• 金额: $ 24.31万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2022-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001890&HistoricalAwards=false
• 关键词: CAREER; Etale; Motivic; Homotopy; Theory

This project lies at the intersection of algebraic topology, algebraic geometry and number theory. It involves applying homotopy theory (a branch of algebraic topology) to study arithmetic and geometry, using coarse aspects or invariants of spaces to study arithmetic phenomena. A generalization of the number of d-dimensional holes in a space is used to control the solutions to certain polynomial equations. When d is equal to one, this has applications to a program of Grothendieck to control solutions using the loops on a space. Maps between certain spaces induce maps between the d-dimensional holes giving rise to a notion of degree. A generalization of degree due to F. Morel is used to study arithmetic properties of singularities. This project furthermore includes the design and implementation of a series of four week-long summer math jobs for gifted high school students from diverse backgrounds. During each of four summers, approximately eight high school students will work on an important mathematical problem which has an elegant or useful known solution, learning the background material as necessary, and then creating learning materials for other students of the same age group. The students will be accompanied by a teacher from their high schools. In addition to the work on the specific mathematical problem, career options in mathematics will be presented and support and mentorship will be provided for students interested in pursuing mathematical careers.Certain arithmetic and geometric phenomena which appear delicate are invariant under appropriate notions of homotopy. Such phenomena motivate the use of homotopy theory to study arithmetic or geometry. The sub-projects contained in this project share the perspective wherein problems in arithmetic or geometry are approached by using Morel-Veoveodky's A1-homotopy theory and applying realization functors. Sub-project 1 studies an enrichment of the Section Conjecture and an approach to proving it. Running the same methods backwards produces results on the differential graded algebra of the absolute Galois group. Sub-project 2 applies an Eilenberg-Moore spectral sequence in étale homotopy to compute (co)homology of branched covers. This has applications to the study of the non-abelian Galois representation given by the fundamental group of the projective line with three points removed, and has applications to sub-project 1. The first step of sub-project 3 is to prove a joint conjecture with Jesse Kass that the quadratic form appearing in the Eisenbud-Levine-Khimshiashvili Signature Formula can be interpreted as a local degree in A1-homotopy, where the natural notion of degree is a quadratic form. We then enrich the Milnor number and use this enrichment to study arithmetic properties of singularities.

该项目在于代数拓扑，代数几何和数字理论的交集。它涉及使用同义理论（代数拓扑结构的一个分支）来研究算术和几何形状，并使用空间的粗糙方面或不变性来研究算术现象。空间中D维孔数量的概括用于控制某些多项式方程的解决方案。当D等于一个时，它将应用于Grothendieck程序，以使用空间上的循环来控制解决方案。某些空间之间的地图影响D维孔之间的地图产生了程度的概念。莫雷尔（F. Morel）引起的程度的概括用于研究奇异性的算术特性。此外，该项目还包括设计和实施一系列为期四个星期的夏季数学工作，为来自潜水员背景的有天赋的高中生。在四个夏季的每个夏季，大约八个高中生都会解决一个重要的数学问题，该问题具有优雅或有用的解决方案，必要时学习背景材料，然后为同龄年龄段的其他学生创建学习材料。学生将陪同他们的高中老师。除了有关特定数学问题的工作外，还将提出数学职业选择，并为有兴趣从事数学职业的学生提供支持和心态。在适当的同质迹象的情况下，确定算术和几何现象似乎是不变的。这种现象促使使用同质理论来研究算术或几何形状。本项目中包含的子项目共享透视图，其中通过使用Morel-Veoveodky的A1-HOMOTOPY理论并应用实现函数来解决算术或几何的问题。亚项目1研究了截面猜想的丰富和证明这一点的方法。向后运行相同的方法在绝对Galois组的差分级代数上产生结果。亚项目2在Étale同拷贝中应用Eilenberg-Moore光谱序列来计算分支盖的（CO）同源性。 This has applications to the study of the non-abelian Galois representation given by the fundamental group of the projective line with three points removed, and has applications to sub-project 1. The first step of sub-project 3 is to prove a joint conjecture with Jesse Kass that the quadratic form appearing in the Eisenbud-Levine-Khimshiashvili Signature Formula can be interpreted as a local degree in a1-homotopy，其中自然概念是二次形式。然后，我们丰富了Milnor数量，并使用此丰富来研究奇异性的算术特性。