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？在这一领域中，如果两个物体中的一个可以连续变形成另一个，则认为它们是相似的。有许多例子可以从几何的角度来处理：著名的是，一个杯子可以连续变形成一个甜甜圈。但是我们只能明确地可视化位于小维度空间中的物体，最多3维。为了研究位于较大维数空间中的物体，我们需要通过代数的透镜来处理这个问题。 代数拓扑学将各种代数不变量赋予几何对象和形状，使得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就可以区分它们。代数拓扑学本身就是一门理论学科，它为理解形状而发展新的不变量，并研究它们的性质。但是，代数拓扑学的工具无论周围空间的维度或物体的大小都同样有效。因此，它们在物理学中有许多应用，因为量子物理和相对论中的问题涉及高维空间中的物体。这些代数不变量也可以应用于大数据分析中的问题。一个大的数据集是一个非常多维的空间中的对象，由于代数拓扑的工具对维度不敏感，它们非常适合展示大数据集的有用属性，这些属性可能很难用经典的统计工具看到。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，这些研究项目涉及色同伦理论，这是一个识别和解释稳定同伦球面群中周期现象的框架。在稳定同伦范畴上引入了一个过滤，并提出了一次研究一个色级和一个素数的问题。目前这个领域的大部分工作都集中在色级2上，素数2是最难的情况。 一个合作项目将计算素数为2的K（2）-局部范畴的Picard群，并使用此信息找到K（2）-局部BrownComenetz对偶的对偶对象。 另一个项目计划在K（2）-局部范畴中使用SpanierWhitehead对偶来证明K（2）-局部球谱的分解结果。另一个合作项目旨在计算色同伦理论中某些环谱上的模块类别的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transchromatic现象。该奖项反映了NSF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.