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.

该奖项支持几何和拓扑研究，更具体地说是几何群体理论领域，该理论研究了对象的几何形状与其对称组的特性之间的联系。 PI将研究组合对象的某些对称性组。此类对象的示例包括树，即节点和边缘的集合，每个都连接一对没有封闭环的节点。其他例子是有限的尺寸多面体复合物，可以将其视为由任意维度的多面体块（例如立方体或四面体）建立的形状。 PI将探索这些对象的几何形状和拓扑之间的联系，其对称组的作用动力学以及对称性的代数结构。该项目还具有教育组成部分。 PI将参与“群体，几何和动态的女性”计划，并将组织工作坊，为这些领域的早期职业女性研究人员提供协作学习和研究经验。其次，PI将在其机构中组织一个受邀演讲者系列，重点关注商业，工业和政府的数学家职业。最后，她将通过组织会议，讲习班，半手，新研究生课程的发展以及一群潜水员的学生群体来继续参与其他专业活动。该项目的第一个研究目标是致力于研究Artin群体在CAT（0）空间的行动的研究，以建立这种行动的僵化。特别是，PI将专注于四个链的编织组，并将其在CAT（0）立方体复合物上进行适当的作用进行分类，并将其适当和共同处理的作用对一般CAT（0）复合物进行分类。在项目的第二部分中，她将调查群体复合物，尤其是Artin群体的复合体的细致方面。在项目的最后一部分中，PI将研究在树木的产物上作用的群体的亚组结构，其目的是最终在此类群体中构建无限地呈现了无限的亚组，建立了他们的不一致。这奖反映了NSF的立法任务，并被认为是通过基金会的智力优点和广泛的评估来进行评估，并值得通过评估来进行评估。