CAREER: Unifying representation stability via Fl-categories

奖项：DMS 1350138，首席研究员：Thomas F. Church 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，或氦 4 原子核凝聚成超流体，而中子会相互抵抗，这是中子星稳定性所必需的。从数学上讲，这种区别可以通过其配置空间的形状来解释，该形状描述了一组无法区分的粒子（例如光子或中子）在空间中配置的所有可能方式。研究人员通过对称性研究构型空间，证明无论粒子在什么空间内移动，构型的拓扑都会稳定，无论存在多少粒子，都会产生相同的行为。受限空间中的配置具有广泛的应用：当电子被限制在场效应晶体管中的二维表面时，配置空间上的新拓扑预测了新型准粒子，这体现在分数量子霍尔效应中；在机器人技术中，工厂车间配置的机器人拓扑描述了一个机器人何时会被其他机器人困住。拟议的项目将把这种稳定性扩展到具有更多对称性的新型配置空间，其稳定性目前无法理解或证明。 研究人员还将开发和教授数学发现实验室，这是一门为本科生提供的基于发现的新课程，旨在培养独立研究的关键技能。 MDL 的学生将研究开放式问题，通过计算机实验和模拟收集数据，提出猜想，并证明有关结果的定理，通过书面报告和讲座展示他们的发现。研究人员的研究重点是表示稳定性，这是一种将表示理论应用于拓扑和代数稳定性问题的技术。表征稳定性非常成功，自三年前研究者引入以来已发表了 20 篇论文。研究者建议通过开展一个长期项目来开发 FI 类别的组合、分类和同源属性，从而扩展和加强表征稳定性的能力。这些为研究各种对称群提供了一个连贯的框架，包括许多使用以前的方法无法处理的对称群。拟议的项目将统一四个不同的稳定性链作为同一理论的各个方面：表示稳定性、同源稳定性、扭曲稳定性和中心稳定性。一个关键的进步是消除了对表示论的依赖，取而代之的是 FI 范畴的组合和同调特性，因此表示稳定性甚至可以应用于积分或模表示。

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.