Equivariant Methods in Chromatic Homotopy Theory

球体是最简单的几何物体之一，它们是更复杂拓扑空间的基石。球的同伦群是考虑到一定变形的球间连续函数的集合。这些群包含拓扑空间之间映射的基本信息，并且与数论、微分拓扑和几何拓扑有很深的联系。然而，尽管它们的定义很简单，球的同伦群却是极难计算的。为了更好地理解这些群，色同伦理论是一个强大的工具，它通过分析光滑单参数形式群的代数几何来组织理论和计算。形式群的模堆具有高度分层，在稳定同伦范畴中对应于Lubin-Tate理论的局域化。Lubin-Tate理论使球的稳定同伦群具有较高的周期性，是色同伦理论研究的重要领域之一。从Hill-Hopkins-Ravenel对Kervaire不变量问题的解决开始，新开发的等变机制为攻克色同伦理论中经典方法难以解决的问题提供了新的方法。计划研究探索等变同伦理论和色同伦理论之间的联系，并使用尖端的等变技术在色同伦理论中产生最先进的计算。主要研究者将通过等变同伦理论和色同伦理论的透镜研究球体稳定同伦群中的周期性现象。在当前和正在进行的项目中，PI建立了色同伦理论中阻碍理论作用与复共轭几何之间的第一个已知联系。利用这个新发现的联系，PI将研究Lubin-Tate理论作为等变光谱，并使用Hill-Hopkins-Ravenel开发的等变机器来计算质数2处的更高色高。本课题旨在通过证明实bordism理论和Lubin-Tate理论的不动点的范数的周期性定理、间隙定理和检测定理，加深等变同伦理论与色同伦理论之间的联系。PI将进行更多的色计算来研究Kervaire不变量问题的最后一个开放情况。PI还将研究Hurewicz图像，证明一般微分模式，并展示Real bordism理论和Lubin-Tate理论在不同群和高度的切片光谱序列中的转色现象。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The spheres are among the simplest geometric objects, and they are the building blocks of more complicated topological spaces. The homotopy groups of spheres are collections of continuous functions between spheres considered up to certain deformations. These groups hold fundamental information about maps between topological spaces and have deep connections to number theory, differential topology, and geometric topology. However, despite their simple definition, the homotopy groups of spheres are extremely difficult to compute. To better understand these groups, chromatic homotopy theory is a powerful tool that organizes theory and computations by analyzing the algebraic geometry of smooth one-parameter formal groups. The moduli stack of formal groups has a stratification by height, which corresponds in the stable homotopy category to localizations with respect to the Lubin-Tate theories. The Lubin-Tate theories give rise to higher periodicity in the stable homotopy groups of spheres, and studying them is one of the most important areas of research in chromatic homotopy theory. Starting from Hill-Hopkins-Ravenel's resolution of the Kervaire invariant problem, the newly developed equivariant machinery offers new methods to attack problems in chromatic homotopy theory that were notoriously difficult to approach via classical methods. The planned research explores the connections between equivariant homotopy theory and chromatic homotopy theory, and uses cutting-edge equivariant technology to produce state-of-the-art computations in chromatic homotopy theory.The principal investigator will study periodicity phenomena in the stable homotopy groups of spheres through the lens of equivariant and chromatic homotopy theory. In current and ongoing projects, the PI establishes the first known connection between the obstruction-theoretic actions in chromatic homotopy theory and the geometry of complex conjugations. Using this newly discovered connection, the PI will study Lubin-Tate theories as equivariant spectra and use equivariant machinery developed by Hill-Hopkins-Ravenel to produce higher chromatic height computations at the prime 2. This project aims to deepen the connection between equivariant and chromatic homotopy theory by proving Periodicity, Gap, and Detection theorems for norms of Real bordism theories and fixed points of Lubin-Tate theories. The PI will carry out more chromatic computations to study the last open case of the Kervaire invariant problem. The PI will also investigate Hurewicz images, prove general differential patterns, and exhibit transchromatic phenomena in the slice spectral sequences of Real bordism theories and Lubin-Tate theories across different groups and heights.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.