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

该项目是代数拓扑学、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，使用空间的粗略方面或不变量来研究算术现象。空间中d维洞个数的推广被用来控制某些多项式方程的解。当d等于1时，这适用于Grothendieck的程序，以使用空间上的循环来控制解。某些空间之间的映射引出了d维洞之间的映射，从而产生了度的概念。利用F.Morel对次数的推广，研究了奇点的算术性质。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。在每四个暑假中，大约有八名高中生会做一道重要的数学题，这道题有一个优雅的或有用的已知解决方案，根据需要学习背景材料，然后为同龄的其他学生制作学习材料。学生们将由他们高中的一名老师陪同。除了特定数学问题的工作外，还将提供数学方面的职业选择，并为有兴趣追求数学职业的学生提供支持和指导。某些看起来微妙的算术和几何现象在适当的同伦概念下是不变的。这种现象促使人们使用同伦理论来研究算术或几何。这个项目中包含的子项目共享这样的观点，即通过使用Morel-Veoveodky的A1-同伦理论和应用实现函数式来解决算术或几何问题。分项目1研究了对该部分猜想的丰富和证明方法。向后运行相同的方法将产生绝对Galois群的微分分次代数的结果。子项目2应用Eilenberg-Moore谱序列的同伦来计算分支覆盖的(余)同调。这对研究去掉三点的射影直线的基本群所给出的非交换Galois表示有一定的应用，对子项目1也有应用。子项目3的第一步是与Jesse Kass证明了一个联合猜想，即出现在Eisenbud-Levine-Khimshiashvili签名公式中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，我们丰富米尔诺数，并利用这种丰富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.