Combinatorics of Manifolds and Stacks with Torus Actions

在该项目下，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。重点是广义设置下的舒伯特微积分，以及具有环面动作的“堆栈”对象。PI还将在图上使用几何和推理统计来尝试通过连接推导神经元分类。总体主题是将强大的几何工具和代数不变量用于几何驱动的组合和图论问题，并允许组合学告知几何结构。该项目还支持PI通过与记者和公众就基本统计和科学推理开展合作，努力提高统计素养和公众教育。这个项目促进了几何学和组合学或计数理论之间的互利关系；一个很好的例子是描述如何计算某些几何空间的交点。具有大量对称性的几何空间可以用组合学中发展出来的方法来描述，而这些描述反过来又可以阐明几何。同样，几何空间激发了对组合学特定方面的探索。该项目促进使用几何和统计技术来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PI担任STATS（统计评估服务，www.stats.org）的研究主任，在那里她与记者合作，研究如何以翔实和诚实的方式向公众展示数学和统计思想。这些努力影响公共政策和立法的形成；他们鼓励公众在数学和定量推理概念方面接受更多的教育，甚至是基本的概念，如相关性和因果关系的区别；他们向公众展示了数学和学术界对社会的重要影响。

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.