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

职业生涯：从等变色同伦理论到物质相：走向边缘的航程

• 批准号: 2143811
• 负责人: Agnes Beaudry
• 金额: $ 47.48万
• 依托单位: University of Colorado at Boulder
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-01 至 2027-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2143811&HistoricalAwards=false
• 关键词: CAREER; Equivariant; Chromatic; Homotopy; Theory

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.

同伦理论研究几何物体的性质，这些物体在连续变形下不变，通过将称为不变量的量与这些物体相关联。一些不变量是稳定的，因为它们独立于某些空间维度的移动。它们更容易计算，因为它们在某种意义上更代数化。该项目有两个主题。研究的第一条线是色同伦理论，这是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：是否存在共享共同属性的稳定不变量族？这些族存在什么样的对称性？如何利用这些对称性来进行显式计算，并学习关于基本几何对象（如高维球体）的新知识？第二条研究路线是与数学家和物理学家的多学科合作的一部分，它使用稳定不变量来研究物质的相位。量子系统是相互作用的粒子的集合，粗略地说，相是一系列量子系统，它们可能在微观上不同，但具有某些宏观性质。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位稳定不变量，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相的分类方面取得进展，以更好地理解材料的基本性质。该项目有一个综合教育部分，其目标之一是通过一系列研究生讲习班使研究生和高年级本科生能够接触这两个研究领域。教育计划还包括本科和研究生研究。特别是，该项目将与博尔德科罗拉多大学的现有倡议合作开展研究，这些倡议致力于促进STEM的多样性，公平性和包容性。一个重要的目标是增加在数学中代表性不足的少数民族的研究的可及性。稳定不变量使用广义上同调理论或更具体地说，称为谱的数学对象进行研究。 色同伦理论的目的是根据它们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。周期性行为与谱的对称性有着密切的关系。这个项目使用等变技术来更好地理解这种关系。它在球面稳定同伦群的研究中有应用。具体来说，该项目探讨了从真实的协边谱建立的Lubin-Tate理论的等变推广的理论和计算特性，这是复协边的等变推广。该项目开发技术来计算从这些理论产生的等变稳定不变量。在另一个方向上，该项目研究了一个猜想，即物质的参数化间隙可逆相形成了广义上同调理论。该项目提出为参数化量子系统构建基态束，并探索等变同伦理论如何为对称系统的研究提供信息。该项目旨在以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，联合收割机结合研究生研讨会与研究会谈，重点是该项目的主要研究领域。这些都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计划结构，在参与者和专家之间创造一个互动和协作的环境。其次，该项目将在拓扑学和物质相方面提供学术学期和夏季本科生研究机会，旨在增加数学历史上被排斥群体的学生获得本科生研究的机会。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。