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。 球面的同伦群是考虑到一定变形的球面之间的连续函数的集合。 这些群包含拓扑空间之间映射的基本信息，与数论、微分拓扑和几何拓扑有着深刻的联系。 然而，尽管定义简单，球面的同伦群却极难计算。 为了更好地理解这些群，色同伦理论是一个强大的工具，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。 形式群的模栈有一个高度分层，这在稳定同伦范畴中对应于鲁宾-泰特理论的局部化。Lubin-Tate理论使球面上的稳定同伦群具有更高的周期性，研究球面上的稳定同伦群是色同伦理论中最重要的研究领域之一。 从Hill-Hopkins-Ravenel对Kervaire不变量问题的解决开始，新开发的等变机器提供了新的方法来攻击色同伦理论中的问题，这些问题是众所周知的难以通过经典方法来解决的。 本研究计划探讨等变同伦理论与色同伦理论之间的联系，并利用最先进的等变技术，在色同伦理论中产生最先进的计算。首席研究员将通过等变和色同伦理论的透镜，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建立了色同伦理论中的障碍理论作用与复共轭几何之间的第一个已知联系。利用这一新发现的联系，PI将把Lubin-Tate理论作为等变谱来研究，并使用Hill-Hopkins-Ravenel开发的等变机制来产生素数2处更高的色高计算。 本项目旨在通过证明真实的协边理论的范数和Lubin-Tate理论的不动点的周期性、间隙和检测定理，加深等变同伦理论和色同伦理论之间的联系。PI将进行更多的色计算，以研究Kervaire不变量问题的最后一个开放案例。PI还将调查Hurewicz图像，证明一般差分模式，并在不同群体和高度的真实的边界理论和Lubin-Tate理论的切片光谱序列中展示transchromatic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价值和更广泛的影响审查标准进行评估而被认为值得支持。

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.