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.

该项目位于代数拓扑、代数几何和数论的交叉点，它涉及应用同伦理论（代数拓扑的一个分支）来研究算术和几何，使用空间的粗方面或不变量来研究算术现象的推广。空间中 d 维孔的数量用于控制某些多项式方程的解，当 d 等于 1 时，这可以应用 Grothendieck 程序来控制。使用空间上的循环的解决方案会导致 d 维孔之间的映射，从而产生度的概念。F. Morel 用于研究奇点的算术属性。为来自不同背景的有天赋的高中生设计和实施一系列为期四个星期的暑期数学作业 在四个暑假中，大约有八名高中生将研究一个重要的数学问题，该问题有一个优雅或有用的已知解决方案。 、学习背景材料必要时，然后为同年龄段的其他学生创建学习材料，除了解决特定数学问题外，还将向学生提供数学职业选择和支持。某些看似微妙的算术和几何现象在适当的同伦概念下是不变的，这些现象激发了使用同伦理论来研究算术或几何。这个项目因此，我们通过使用 Morel-Veoveodky 的 A1 同伦理论和应用实现函子来解决算术或几何中的问题，子项目 1 研究了截面猜想的丰富以及向后证明它的方法。关于绝对伽罗瓦群的微分分级代数 子项目 2 应用 etale 同伦中的 Eilenberg-Moore 谱序列来计算 的（共）同调。这适用于研究由删除三点的射影线的基本群给出的非阿贝尔伽罗瓦表示，并且适用于子项目 1。子项目 3 的第一步是证明与 Jesse Kass 共同猜想，Eisenbud-Levine-Khimshiashvili 签名公式中出现的二次形式可以解释为 A1 同伦中的局部度，其中度的自然概念是然后我们丰富米尔诺数并使用这种丰富来研究奇点的算术性质。