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

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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.