Picard Groups and Duality in Chromatic Homotopy Theory at the Prime 2.

代数拓扑学研究物体的形状，旨在回答这样一个问题：我们如何判断两个物体是否相似？在这个场中，如果两个物体中的一个可以连续地变形成另一个，则认为它们是相似的。有很多例子可以从几何的角度来看：著名的，一个马克杯可以不断地变形成一个甜甜圈。但我们只能明确地可视化小维度空间中的物体，甚至是三维空间。为了研究更大维度空间中的物体，我们需要从代数的角度来研究这个问题。代数拓扑学将各种代数不变量赋给几何对象和形状，使相似的对象具有相同的不变量。然后，给定两个对象，我们只需要计算它们的不变量，以便能够区分它们。代数拓扑学本身就是一门理论学科，它发展了新的不变量来理解形状，并研究了它们的性质。但是代数拓扑的工具不管环境空间的维度，或者对象的大小，都能很好地工作。因此，它们在物理学中有许多应用，因为量子物理学和相对论中的问题处理的是高维空间中的物体。这些代数不变量也可以应用于大数据分析中的问题。一个大数据集是一个对象在非常多的维度空间，由于代数拓扑的工具是不敏感的维度，他们非常适合展示大数据集的有用的性质，可能很难看到与经典统计工具。该奖项的更广泛影响包括组织研讨会和促进妇女参与数学的活动。这些研究项目涉及到色同伦理论，它是识别和解释稳定球同伦群中周期现象的一个框架。引入了稳定同伦范畴上的滤波，提出了一次研究一个色阶和一个素数的问题。目前该领域的大部分工作都集中在音阶2上，其中素数2是最难的。一个合作项目将计算K(2)-局部范畴在质数2处的Picard群，并利用该信息找到K(2)-局部BrownComenetz对偶的对偶对象。另一个项目计划使用K(2)-局部范畴中的西班牙人-怀特黑德对偶来证明K(2)-局部球谱的分解结果。一个额外的合作努力旨在计算某些环谱上模类的Picard群，计算感兴趣的K(2)局部谱的同伦群，并研究GrossHopkins对偶中的转色现象。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。

Algebraic topology studies shapes of objects and aims to answer the question: how can we tell whether two objects are similar? In this field two objects are considered to be similar if one of them can be continuously deformed into the other. There are many examples which can be approached from the geometric point of view: famously, a mug can be continuously deformed into a donut. But we are only able to explicitly visualize objects which lie in spaces of small dimension, up to 3 dimensions. In order to study objects lying in spaces of larger dimension we need to approach the problem through the lens of algebra. Algebraic topology assigns various algebraic invariants to geometric objects and shapes in such a way that similar objects will have the same invariant. Then, given two objects, we only need to compute their invariants in order to be able to distinguish them. Algebraic topology itself is a theoretical subject which develops new such invariants for understanding shapes, and studies their properties. But the tools of algebraic topology work equally well regardless of the dimension of the ambient space, or the size of the objects. Due to this, they have numerous applications in physics, since problems in quantum physics and relativity deal with objects in spaces of high dimension. These algebraic invariants can also be applied to problems in the analysis of large data. A large data set is an object in a space of very many dimensions, and since tools of algebraic topology are insensitive to dimension, they are well suited to exhibit useful properties of large data sets which might be difficult to see with classical statistical tools. Broader impacts under this award include seminar organization and events promoting the participation of women in mathematics.These research projects concern chromatic homotopy theory, which is a framework for identifying and explaining periodic phenomena in stable homotopy groups of spheres. It introduces a filtration on the stable homotopy category, and proposes to study the problem one chromatic level and one prime at a time. Most of the current work in this field is focused on chromatic level 2, with prime 2 being the hardest case. A collaborative project will compute the Picard group of the K(2)-local category at the prime 2 and use this information to find the dualizing object for the K(2)-local BrownComenetz duality. Another project plans to use SpanierWhitehead duality in the K(2)-local category to prove a decomposition result for the K(2)-local sphere spectrum. An additional collaborative effort aims to compute the Picard groups of categories of modules over certain ring spectra in chromatic homotopy theory, computing the homotopy groups of K(2)-local spectra of interest and studying the transchromatic phenomena in GrossHopkins duality.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.