CAREER: Groups Acting on Combinatorial Objects

职业：作用于组合对象的团体

• 批准号: 2238198
• 负责人: Katarzyna Jankiewicz
• 金额: $ 55万
• 依托单位: University of California-Santa Cruz
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2028-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2238198&HistoricalAwards=false
• 关键词: CAREER; Groups; Acting; Combinatorial; Objects

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.

该奖项支持几何学和拓扑学的研究，更具体地说，是几何群论领域的研究，该领域研究对象的几何和它的对称性群的性质之间的联系。PI将研究组合对象的某些对称性。这种对象的例子包括树，即节点和边的集合，每个树连接不包含闭合环的一对节点。其他例子是有限维的多面体复合体，它可以被认为是由任意维度的多面体块组成的形状，如立方体或四面体。PI将探索这些对象的几何和拓扑之间的联系，它们的对称群的作用的动力学，以及对称的代数结构。该项目还包括教育内容。该协会将参与“妇女在群体、几何学和动力学中”的倡议，并将为这些领域的早期职业女性研究人员举办促进协作学习和研究经验的讲习班。其次，PI将在她的机构组织一系列特邀演讲者，重点关注商业、工业和政府中数学家的各种职业。最后，她将继续参与其他专业活动，通过组织会议、研讨会、研讨会、开发新的研究生课程以及指导不同的学生群体。该项目的第一个研究目标致力于研究Artin小组在CAT(0)空间上的行动，目的是建立此类行动的刚性。特别是，PI将聚焦于四条链上的辫子基团，并对其对CAT(0)立方体络合物的适当作用进行分类，以及对一般CAT(0)络合物的固有作用和余紧作用进行分类。在项目的第二部分，她将研究群的复合体的无限方面，特别是Artin群。在项目的最后部分，PI将研究作用在树的乘积上的群的子群结构，目标是在这样的群中构造有限生成的无限呈现的子群，建立它们的无连贯性。这一奖项反映了NSF的法定使命，并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，被认为值得支持。