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

本课题是代数拓扑学、代数几何和数论的交叉课题。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，使用空间的粗糙方面或不变量来研究算术现象。利用空间中d维空穴数量的一般化来控制某些多项式方程的解。当d = 1时，可应用于格罗滕迪克程序，利用空间上的循环控制解。特定空间之间的映射导致了d维空穴之间的映射，从而产生了度的概念。利用F. Morel的广义度来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作业。在四个暑假的每一个暑假里，大约八名高中生将研究一个重要的数学问题，这个问题有一个优雅或有用的已知解，学习必要的背景材料，然后为同年龄组的其他学生创建学习材料。学生们将由一名高中老师陪同。除了研究具体的数学问题外，还将提供数学方面的职业选择，并为有兴趣从事数学职业的学生提供支持和指导。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。这种现象促使我们使用同伦理论来研究算术或几何。本项目包含的子项目分享了使用Morel-Veoveodky的a1 -同伦理论和应用实现函子来解决算术或几何问题的观点。子课题1研究了剖面猜想的丰富和证明方法。将相同的方法反向运行，可以在绝对伽罗瓦群的微分梯度代数上得到结果。子项目2应用一个在<s:1>同伦上的Eilenberg-Moore谱序列计算分支复盖的（co）同伦。这适用于研究消去三点的射影线的基群给出的非阿贝尔伽罗瓦表示，也适用于子项目1。子项目3的第一步是证明与Jesse Kass的一个联合猜想，即Eisenbud-Levine-Khimshiashvili签名公式中出现的二次型可以解释为a1 -同伦中的局部次，其中次的自然概念是二次型。然后对米尔诺数进行充实，并利用这种充实来研究奇点的算术性质。

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.