LEAPS-MPS: Algorithms and Geometry in Group Theory

该奖项的全部或部分资金来自《2021年美国救援计划法案》(公法117-2)。任何几何对象的对称性都编码在称为群的代数结构中。几何群论的主要目标之一是在物体的几何和它的对称群的代数之间建立联系。这导致了一种将拓扑学、代数、组合学和计算机科学联系起来的强大方式。这个项目追求的一个感兴趣的领域将是寻找具有有趣的几何属性的组，这些组在计算上不是“简单的”。PI还将研究直角Coxeter群，这是近100年前引入的一类反射群，使用概率和组合学的技术来分析它们的几何和代数。作为该项目的一部分，PI将为巴德高中早期学院的学生组织和现场直播数学家的讲座，通过将他们与该领域发生的有趣工作联系起来，为来自城市和郊区未被充分代表的群体的学生提供机会，并了解本科生和研究生水平的数学研究途径。该项目涉及三个与几何和算法问题相关的小组研究方向。PI将研究直角Coxeter群的不同方面：一方面将利用随机图的ErdöS-Renyi模型来研究随机直角Coxeter群的负曲率性质；另一方面将探索直角Coxeter群的外自同构群的拓扑有限性和负曲率性质。PI还将研究CAT(0)组的新实例的自动性，并研究可自动堆叠的组的语言理论属性。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). The symmetries of any geometric object are encoded in an algebraic structure called a group. One of the primary goals of geometric group theory is to draw connections between the geometry of an object and the algebra of its symmetry group. This leads to a powerful way to link topology, algebra, combinatorics and computer science. One area of interest pursued in this project will be finding groups with interesting geometric properties that fail to be computationally “simple.” The PI will also study right-angled Coxeter groups, a class of reflection groups introduced almost 100 years ago, using techniques from probability and combinatorics to analyze their geometry and algebra. As part of this project, the PI will organize and live-stream talks by mathematicians to students at the Bard High School Early Colleges to provide opportunities to students from underrepresented groups in urban and suburban areas by connecting them with interesting work happening in the field as well learn about pathways to mathematical research on the undergraduate and post-graduate level.This project involves three research directions tied to geometric and algorithmic problems in groups. The PI will study different aspects of right-angled Coxeter groups: in one direction the Erdös-Renyi model of random graphs will be used to study negative curvature properties of random right-angled Coxeter groups; in another the topological finiteness and negative curvature properties of outer automorphism groups of right-angled Coxeter groups will be explored. The PI will also study automaticity for new examples of CAT(0) groups, and study language theoretic properties of autostackable groups.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.