CAREER: From Equivariant Chromatic Homotopy Theory to Phases of Matter: Voyage to the Edge

同伦理论通过将称为不变量的量与连续变形下不变的几何物体联系起来，研究这些物体的性质。有些不变量是稳定的，因为它们不依赖于某些空间维度的变化。它们更容易计算因为在某种意义上它们更具有代数性质。这个项目有两个主题。研究的第一条线是在色同伦理论，一个研究稳定不变量的结构性质的数学领域。该项目的这一部分旨在回答以下问题：是否存在具有共同属性的稳定不变量族？这些家庭存在什么样的对称性？如何利用这些对称性进行显式计算，并学习基本几何对象（如高维球体）的新知识？第二项研究是数学家和物理学家多学科合作的一部分，它使用稳定不变量来研究物质的相。量子系统是相互作用的粒子的集合，粗略地说，相是一组量子系统，它们在微观上可能不同，但具有某些宏观特性。对于某些类型的量子系统，相位类型可以通过稳定不变量来检测。本项目旨在构建新的相稳定不变量，并研究具有一定对称性的量子系统的稳定不变量。更广泛的目标是在物质的相分类方面取得进展，以便更好地了解材料的基本特性。该项目有一个综合的教育组成部分，其中一个目标是通过一系列研究生讲习班使研究生和高级本科生能够接触到这两个研究领域。该教育计划还包括本科生和研究生的研究。特别是，该项目将与科罗拉多大学博尔德分校的现有项目合作开展研究，这些项目致力于促进STEM的多样性、公平性和包容性。一个重要的目标是增加数学领域代表性不足的少数群体获得研究的机会。稳定不变量是用广义上同调理论来研究的，或者更具体地说，用称为谱的数学对象来研究。色同伦理论旨在根据谱族计算的稳定不变量所表现出的不同周期行为对谱族进行分类。周期行为与谱的对称性有密切的关系。本项目使用等变技术来更好地理解这种关系。它在球的稳定同伦群的研究中也有应用。具体来说，该项目探索了Lubin-Tate理论的等变推广的理论和计算性质，这些理论是建立在实bordism谱上的，这是复bordism的等变推广。该项目开发了计算由这些理论产生的等变稳定不变量的技术。在一个不同的方向，该项目检验了一个猜想，即参数化的间隙可逆相的物质形成一个广义上同调理论。该项目提出了参数化量子系统的基态束的构建，并探索了等变同伦理论如何能够为对称性系统的研究提供信息。该项目旨在通过两种方式增加学生对这些主题的了解。首先，该项目包括两个为期五天的数学活动，将研究生研讨会与研究会谈结合起来，重点关注项目的主要研究领域。这些将是一个正在进行的系列的一部分，将继续超过五年的项目期限。这些活动将通过将主动学习纳入项目结构，在参与者和专家之间创造一个互动和协作的环境。其次，该项目将在学期和夏季为本科生提供拓扑和物质相的研究机会，目的是增加历史上被数学排斥群体的学生获得本科研究的机会。该项目还将支持从事多元化、公平和包容倡议的研究生。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。

Homotopy theory studies properties of geometric objects that are unchanged under continuous deformations by associating quantities called invariants to these objects. Some invariants are stable in the sense that they are independent of certain spatial dimension shifts. They are easier to compute because they are in some sense more algebraic. This project has two main themes. The first line of investigation is in chromatic homotopy theory, a field of mathematics that studies structural properties of stable invariants. This part of the project seeks to answer questions such as: Are there families of stable invariants that share common properties? What kind of symmetries exist for these families? How can these symmetries be used to do explicit computations and learn new things about fundamental geometric objects such as higher dimensional spheres? The second line of investigation is part of a multi-disciplinary collaboration with mathematicians and physicists, which uses stable invariants to study the phase of matter. A quantum system is a collection of interacting particles and, roughly, a phase is a family of quantum systems that may be different microscopically, but share certain macroscopic properties. For certain types of quantum systems, the phase type can be detected by stable invariants. This project aims to construct new stable invariants of phases and to study stable invariants of quantum systems equipped with certain symmetries. The broader goal is to make progress on the classification of phases of matter to better understand the fundamental properties of materials. The project has an integrated educational component, one goal of which is to make the two areas of research accessible to graduate students and advanced undergraduates through a series of graduate workshops. The educational plan also includes undergraduate and graduate research. In particular, the project will conduct research in collaboration with existing initiatives at the University of Colorado, Boulder that work to promote diversity, equity and inclusion in STEM. An important goal is to increase the accessibility of research for underrepresented minorities in mathematics.Stable invariants are studied using generalized cohomology theories or, more specifically, mathematical objects called spectra. Chromatic homotopy theory aims to classify families of spectra according to different periodic behaviors exhibited by the stable invariants they compute. There is a close relationship between periodic behaviors and the symmetries of the spectrum. This project uses equivariant techniques to better understand this relationship. It has applications to the study of stable homotopy groups of spheres. Specifically, the project explores theoretical and computational properties of equivariant generalizations of Lubin-Tate theories that are built from the Real bordism spectrum, an equivariant generalization of complex bordism. The project develops techniques to compute the equivariant stable invariants arising from these theories. In a different direction, the project examines a conjecture that parametrized gapped invertible phases of matter form a generalized cohomology theory. The project proposes the construction of a ground-state bundle for parametrized quantum systems and explores how equivariant homotopy theory can inform the study of systems with symmetries. The project aims to increase accessibility of these topics to students in two ways. First, the project includes two five days mathematical events that combine a graduate workshop with research talks with a focus on the main areas of research of the project. These are to be part of an ongoing series that will continue beyond the five year duration of the project. These events will create an interactive and collaborative environment between participants and experts by incorporating active-learning in the program structure. Secondly, the project will engage in academic term and summer undergraduate research opportunities in topology and phases of matter with a goal to increase access to undergraduate research for students from historically excluded groups in mathematics. The project will also support graduate students working on diversity, equity and inclusion initiatives.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.