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

这个项目位于代数拓扑，代数几何和数论的交叉点。它涉及应用同伦理论（代数拓扑学的一个分支）来研究算术和几何，使用粗糙的方面或空间的不变量来研究算术现象。 空间中d维洞的数目的推广被用来控制某些多项式方程的解。当d等于1，这有应用程序的程序格罗滕迪克控制解决方案使用的循环空间。 某些空间之间的映射诱导了d维洞之间的映射，从而产生了度的概念。推广了F.莫瑞尔用于研究奇点的算术性质。该项目还包括为来自不同背景的天才高中生设计和实施一系列为期四周的暑期数学工作。在四个夏天的每一个，大约八名高中生将工作的一个重要的数学问题，有一个优雅的或有用的已知解决方案，学习必要的背景材料，然后创建学习材料，为其他学生的同龄组。学生们将由他们高中的一名老师陪同。除了具体的数学问题的工作，在数学的职业选择将被提出，并支持和指导将提供有兴趣追求数学生涯的学生。某些算术和几何现象，似乎微妙的同伦的适当概念下不变。这些现象激发了同伦理论在算术或几何中的应用。在这个项目中包含的子项目共享的角度，其中问题的算术或几何接近使用Morel-Veelodky的A1-同伦理论和应用实现函子。子项目1研究了截面猜想的一个扩充及其证明方法。反向运行相同的方法，得到关于绝对伽罗瓦群的微分分次代数的结果。子项目2应用Eilenberg-Moore谱序列在étale同伦中计算分支覆盖的（上）同调。这有应用程序的非阿贝尔伽罗瓦表示的研究所给予的基本组的射影线与三个点删除，并已应用到子项目1。子项目3的第一步是证明与Jesse Kass的联合猜想，即Eisenbud-Levine-Khimshiashvili签名公式中出现的二次型可以解释为A1-同伦中的局部度，其中度的自然概念是二次型。然后，我们丰富的Milnor数，并使用这种丰富的研究算术性质的奇点。

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.