CAREER: Unifying representation stability via Fl-categories

奖项：DMS 1350138，主要研究者：托马斯F.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核可以凝聚成超流体，而中子则相互抵制，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组态空间的形状来解释，它描述了一组不可区分的粒子（如光子或中子）在空间中的所有可能配置方式。研究者通过对称性研究组态空间，证明无论粒子在什么空间内移动，组态的拓扑结构都将稳定，无论存在多少粒子，都会产生相同的行为。受限空间中的拓扑有着广泛的应用：当电子被限制在二维表面（如场效应晶体管）时，位形空间上的新拓扑预测了新类型的准粒子，这表现在分数量子霍尔效应中;在机器人技术中，工厂车间中配置的机器人的拓扑描述了一个机器人何时可以被其他机器人捕获。拟议的项目将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。 研究员还将开发和教授数学发现实验室，这是一门面向本科生的新的基于发现的课程，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，制定图表，并证明其结果的定理，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，自三年前由研究人员引入以来，已经发表了20多篇论文。研究者建议通过开展一个长期项目来发展FI-范畴的组合、范畴和同调性质，以扩展和加强表征稳定性的能力。这些为研究各种各样的对称群提供了一个连贯的框架，其中包括许多无法使用以前的方法处理的对称群。该项目将把稳定性的四个不同方面统一为同一理论的各个方面：表征稳定性、同调稳定性、扭曲稳定性和中心稳定性。一个关键的进步是对表示论的依赖被去除，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。

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.