CAREER: Unifying representation stability via Fl-categories

职业：通过 Fl 类别统一表示稳定性

• 批准号: 1350138
• 负责人: Thomas Church
• 金额: $ 43.31万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1350138&HistoricalAwards=false
• 关键词: CAREER; Unifying; representation; stability; via

Award: DMS 1350138, Principal Investigator: Thomas F. ChurchThe physical distinction between bosons and fermions explains why photons can cohere into a laser, or helium-4 nuclei into a superfluid, whereas neutrons resist each other, as is necessary for the stability of neutron stars. Mathematically, this distinction is explained by the shape of their configuration space, which describes all possible ways that a collection of indistinguishable particles (such as photons or neutrons) can be configured in space. The investigator studies configuration spaces via their symmetries, proving that no matter what space the particles move within, the topology of configurations will stabilize, yielding identical behavior no matter how many particles are present. Configurations in restricted spaces have widespread applications: when electrons are confined to 2-dimensional surfaces as in field-effect transistors, new topology on the configuration space predicts new types of quasiparticles, which manifest in the fractional quantum Hall effect; in robotics, the topology of robots configured on a factory floor describes when one robot can be trapped by others. The proposed project would extend this stability to new types of configuration spaces possessing even more symmetry, whose stability cannot currently be understood or proved. The investigator will also develop and teach Math Discovery Lab, a new discovery-based course for undergraduates which develops critical skills in independent research. Students in MDL will investigate open-ended problems, collect data via computer experimentation and simulation, formulate conjectures, and prove theorems about their results, presenting their findings through written reports and lectures.The investigator's research focuses on representation stability, a technique applying representation theory to stability problems in topology and algebra. Representation stability has been very successful, leading to 20+ papers since its introduction by the investigator three years ago. The investigator proposes to extend and strengthen the power of representation stability by undertaking a long-term project to develop the combinatorial, categorical, and homological properties of FI-categories. These provide a coherent framework for studying a wide variety of symmetry groups, including many which could not be handled using previous methods. The proposed project will unify four distinct strands of stability as facets of the same theory: representation stability, homological stability, twisted stability, and central stability. A key advance is that the dependence on representation theory is removed, replaced instead by the combinatorial and homological properties of FI-categories, so that representation stability can be applied even to integral or modular representations.

玻色子和费米子之间的物理区别解释了为什么光子可以凝聚成激光，或者氦-4原子核可以凝聚成超流体，而中子相互抵抗，这是中子星稳定所必需的。从数学上讲，这种区别可以用它们的构型空间的形状来解释，构型空间描述了一组不可区分的粒子（如光子或中子）在空间中的构型的所有可能方式。研究者通过对称来研究构型空间，证明无论粒子在哪个空间运动，构型的拓扑结构都会稳定，无论存在多少粒子都会产生相同的行为。受限空间中的构型具有广泛的应用：当电子被限制在二维表面（如场效应晶体管）时，构型空间上的新拓扑预测了新型准粒子，这些准粒子表现为分数量子霍尔效应；在机器人技术中，在工厂车间配置的机器人拓扑结构描述了一个机器人何时可能被其他机器人困住。提议的项目将这种稳定性扩展到具有更多对称性的新型构型空间，其稳定性目前无法理解或证明。该研究员还将开发和教授数学发现实验室，这是一门针对本科生的基于发现的新课程，旨在培养独立研究的关键技能。MDL的学生将调查开放式问题，通过计算机实验和模拟收集数据，制定推测，并证明有关其结果的定理，并通过书面报告和讲座展示他们的发现。研究者的研究重点是表征稳定性，一种将表征理论应用于拓扑和代数稳定性问题的技术。代表性稳定性非常成功，自三年前研究者引入以来，已经发表了20多篇论文。研究者建议通过开展一个长期的项目来发展FI-categories的组合性、分类性和同构性，以扩展和加强表征稳定性的力量。这为研究各种各样的对称群提供了一个连贯的框架，包括许多用以前的方法无法处理的对称群。拟议的项目将统一四种不同的稳定性，作为同一理论的各个方面：表征稳定性、同源稳定性、扭曲稳定性和中心稳定性。一个关键的进步是对表示理论的依赖被移除，取而代之的是fi -范畴的组合和同调性质，因此表示稳定性甚至可以应用于积分或模表示。