CAREER: Groups Acting on Combinatorial Objects

该奖项支持几何和拓扑学的研究，更具体地说，在几何群论领域，它研究对象的几何形状之间的联系，以及其对称性组的属性。PI将研究组合对象的某些对称群。这样的对象的示例包括树，即节点和边的集合，每个节点和边连接一对节点，其不包含闭环。其他例子是有限维多面体复合体，可以认为是由任意尺寸的多面体块构建的形状，例如立方体或四面体。PI将探索这些对象的几何和拓扑结构之间的联系，它们的对称群的动态作用，以及对称的代数结构。该项目还包括教育部分。公共研究所将参与“妇女参与群体、几何学和动力学”倡议，并将组织讲习班，促进这些领域的早期职业女性研究人员的协作学习和研究经验。其次，PI将在她的机构组织一个特邀演讲者系列，重点介绍数学家在商业，工业和政府中的各种职业生涯。最后，她将继续从事其他专业活动，通过组织会议，研讨会，研讨会，开发新的研究生课程，并指导不同的学生群体。该项目的第一个研究目标是致力于研究Artin群体在CAT（0）空间上的行为，目标是建立这种行为的刚性。特别地，PI将专注于四股辫群，并将其在CAT（0）立方体复形上的适当作用分类，以及将其在一般CAT（0）复形上的适当作用和上紧作用分类。在该项目的第二部分，她将调查的profinite方面的复杂的群体，特别是阿廷集团。在该项目的最后一部分，PI将研究几何上作用于树的产品上的群的子群结构，目标是在这些群中构建无限生成的无限呈现的子群，建立它们的不一致性。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

This award supports research in geometry and topology, and more specifically in the area of geometric group theory, which studies the connection between the geometry of an object, and the properties of the group of its symmetries. The PI will study certain groups of symmetries of combinatorial objects. Examples of such objects include trees, i.e. collections of nodes and edges, each connecting a pair of nodes, which contain no closed loops. Other examples are finite dimensional polyhedral complexes, which can be thought of as shapes built out of polyhedral blocks of arbitrary dimension, such as cubes or tetrahedra. The PI will explore the connection between the geometry and topology of those objects, the dynamics of the action of their symmetry groups, and the algebraic structure of the symmetries. The project also has educational components. The PI will be involved in the initiative “Women in Groups, Geometry, and Dynamics” and will organize workshops facilitating collaborative learning and research experience for early career female researchers in those areas. Secondly, the PI will organize an invited speaker series at her institution, focusing on various careers of mathematicians in business, industry, and government. Finally, she will continue her engagement in other professional activities, through organization of conferences, workshops, seminars, development of a new graduate course, and mentorship of a diverse group of students.The first research goal of the project is devoted to the study of actions of Artin groups on CAT(0) spaces with the goal of establishing a rigidity of such actions. In particular, the PI will focus on the braid group on four strands, and classify its proper actions on CAT(0) cube complexes, as well as classify its proper and cocompact actions on general CAT(0) complexes. In the second part of the project, she will investigate the profinite aspects of complexes of groups, and in particular of Artin groups. In the final part of the project, the PI will study the subgroup structure of groups acting geometrically on a product of trees, with the goal of constructing finitely generated infinitely presented subgroups in such groups, establishing their incoherence.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.