New Tools in Chromatic Homotopy Theory

现代数学的大部分内容都与数学内部不同领域之间错综复杂的相互作用有关。这个项目的主要目标之一是使用逻辑的想法，以比较拓扑和代数。拓扑学中的一个领域叫做色同伦理论，它通过把空间“分解”成素部分来研究空间，就像整数有一个素分解一样。使用逻辑的思想，我们可以理解素数空间的集合，因为素数趋于无穷大。事实证明，由此产生的空间集合可以完全使用代数来理解。这是一个很新的想法，我们希望把它发展成一个完整的理论。这将允许纯代数的结果有重要的拓扑后果。 这个项目的另一个主要目标是使用拓扑学在几何和代数之间建立一座新的桥梁。这对数学家来说是一个熟悉的故事。经典上，某些称为向量丛的几何对象被用来产生空间的一个重要代数不变量。这个代数不变量在80年代被推广，但在推广的过程中，与几何的联系有些丢失。另一方面，这种推广与代数中的一个领域--算术几何--有着很好的关系。进一步发展这种关系与算术几何应该暴露部分的几何是losed.The PI计划开发新的工具，在色同伦理论，提供概念和计算的洞察力，同时揭示色同伦理论的支持几何和算术几何之间的桥梁。这些工具包括一个几何结构的摩拉瓦E-理论方面的Stolz-Teichner领域的理论，描述的渐近行为的色同伦理论，介绍德林费尔德椭圆模块到色计算，和融合系统的组合描述的分类空间的有限群时，本地化的色素数导致决议的猜想的拉文埃尔的。这些工具都是建立在从PI的工作和transchromatic同伦理论的应用获得的见解。正因为如此，PI也将继续发展transchromatic同伦理论。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

Much of modern mathematics is concerned with the intricate interactions between different areas inside of mathematics. One of the primary goals of this project is to use ideas from logic in order to compare topology and algebra. An area of topology called chromatic homotopy theory studies spaces by ``factoring" them into prime parts in the same way that an integer has a prime factorization. Using ideas from logic, we can understand the collection of prime spaces as the prime tends to infinity. It turns out that the resulting collection of spaces can be completely understood using only algebra. This idea is quite new and we hope to develop it into a full theory. This will allow for purely algebraic results to have important topological consequences. Another primary goal of this project is to use topology to build a new bridge between geometry and algebra. This is a familiar story to mathematicians. Classically, certain geometric objects called vector bundles were used to produce an important algebraic invariant of spaces. This algebraic invariant was generalized in the 80's, but in the process of generalization the connection to geometry was somewhat lost. On the other hand, the generalization has a beautiful relationship to an area of algebra called arithmetic geometry. Further developing this relationship with arithmetic geometry should expose part of the geometry that was lost.The PI plans to develop new tools in chromatic homotopy theory that provide both conceptual and computational insight while revealing chromatic homotopy theory as the support for a bridge between geometry and arithmetic geometry. These tools include a geometric construction of Morava E-theory in terms of Stolz--Teichner field theories, a description of the asymptotic behavior of chromatic homotopy theory that introduces Drinfeld elliptic modules into chromatic calculations, and a fusion-system-like combinatorial description of the classifying spaces of finite groups when localized at a chromatic prime leading to the resolution of a conjecture of Ravenel's. These tools are all built on insights gained from the PI's work on and applications of transchromatic homotopy theory. Because of this, the PI will also continue to develop transchromatic homotopy theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.