Combinatorics of Manifolds and Stacks with Torus Actions

在这个项目中，首席研究员(PI)将开发具有李群作用的几何空间(流形和光滑堆栈)的组合学，并在一个补充程序中探索组合学如何约束几何对象及其分类。重点放在广义环境中的舒伯特微积分，以及具有圆环作用的“堆叠”对象。PI还将使用图形的几何和推断统计数据，试图通过连接得出神经元分类。主要的主题是利用强大的几何工具和代数不变量来处理几何激励的组合和图论问题，并允许组合学告知几何结构。该项目还通过她与记者和公众在基本统计和科学推理方面的合作，支持PI努力提高统计素养和公共教育。这个项目促进了几何学和组合学之间的互惠关系，或计数理论；一个很好的例子是描述如何计算某些几何空间的交点。具有很大对称性的几何空间可以用组合学中发展起来的方法来描述，这些描述反过来可以阐明几何。同样，几何空间激发了对组合学特定方面的探索。该项目促进了几何和统计技术的使用，以研究神经科学中的一个重要问题：从图论的角度来看，我们的大脑是如何组织的。最后，该项目支持国际统计研究所(STATS)(统计评估处，www.stats.org)担任研究主任，在那里她与记者合作，如何以信息丰富和诚实的方式向公众介绍数学和统计思想。这些努力影响了公共政策和立法的形成；它们鼓励公众更多地了解数学和定量推理概念，甚至是基本概念，如相关性和因果关系之间的区别；它们向公众表明，数学和学术界对社会具有重要影响。

Under this project, the principal investigator (PI) will develop the combinatorics for geometric spaces (manifolds and smooth stacks) with Lie group actions and, in a complimentary program, explore how the combinatorics constrain the geometric objects and their classification. The focus is on Schubert calculus in a generalized setting, and 'stacky' objects with torus actions. The PI will also use geometry and inferential statistics on graphs to try to derive neuronal classification through connectivity. The overarching theme is to bring powerful geometric tools and algebraic invariants to bear on geometrically motivated combinatorial and graph theoretic questions, and to allow the combinatorics to inform the geometric structures. The project also supports the PI's efforts to increase statistical literacy and public education through her work with journalists and the public on basic statistical and scientific reasoning. This project promotes the mutual beneficial relationship between geometry and combinatorics, or the theory of counting; an excellent example is describing how to count intersections of certain geometric spaces. Geometric spaces with a lot of symmetry can be described using methods developed in combinatorics, and these descriptions can in turn shed light on the geometry. Similarly, geometric spaces motivate the exploration of specific aspects of combinatorics. The project promotes the use of geometric and statistical techniques to study an important question in neuroscience: how our brains are organized from a graph-theoretic point of view. Finally, the project supports the PI as Director of Research at STATS (Statistical Assessment Service, www.stats.org), where she works with journalists on how to present mathematical and statistical ideas to the public in an informative and honest way. These efforts impact how public policy and legislation is formed; they encourages the public to become more educated regarding mathematics and quantitative reasoning concepts, even basic ones such as the difference between correlation and causation; and they show the public that mathematics and academia have an important impact on society.