Combinatorics of Manifolds and Stacks with Torus Actions

George Mason University, Fairfax VA

Investigators

Abstract

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.