Equivariant Methods in Chromatic Homotopy Theory

球体是最简单的几何对象之一，它们是更复杂的拓扑空间的构建块。球面的同伦群是被认为达到一定变形的球面之间的连续函数的集合。这些群包含关于拓扑空间之间映射的基本信息，并且与数论、微分拓扑学和几何拓扑学有着深刻的联系。然而，尽管球面的同伦群的定义很简单，但它们的同伦群却很难计算。为了更好地理解这些群，色同伦理论是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的有力工具。形式群的模堆叠具有高度分层，这在稳定同伦范畴中对应于关于Lubin-Tate理论的局部化。Lubin-Tate理论在球面的稳定同伦群中产生了较高的周期性，研究球面稳定同伦群是色同伦论研究的重要领域之一。从Hill-Hopkins-Ravenel对Kervaire不变量问题的解决出发，新发展的等变机器为解决色同伦理论中的问题提供了新的方法，这些问题是出了名的难以用经典方法来解决的。计划中的研究探索等变同伦理论与色同伦理论之间的联系，并利用前沿的等变技术产生最先进的色同伦理论计算。主要研究人员将通过等变理论和色同伦理论的透镜来研究球面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在色同伦理论中的阻塞理论作用和复共轭几何之间建立了第一个已知的联系。利用这个新发现的联系，PI将把Lubin-Tate理论作为等变谱来研究，并使用Hill-Hopkins-Ravenel开发的等变机制在素数2产生更高的色高度计算。本项目旨在通过证明实边界理论的范数和Lubin-Tate理论的不动点的周期性、间隙和检测定理来深化等变和色同伦理论之间的联系。PI将进行更多的色度计算来研究Kervaire不变量问题的最后一个公开情况。PI还将研究Hurewicz图像，证明一般差异模式，并在不同组和不同高度的Real Bordism理论和Lubin-Tate理论的切片光谱序列中展示变色现象。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.