Constructing Counterexamples in Group Rings and Algebraic Topology

群环和代数拓扑中构造反例

• 批准号: EP/V047604/1
• 负责人: David Craven
• 金额: $ 25.77万
• 依托单位: University of Birmingham
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FV047604%2F1
• 关键词: Constructing; Counterexamples; Group; Rings; Algebraic

Groups are the mathematical embodiment of the concept of symmetries of an object, and representation theory studies how these symmetries act on space. The representations of a group are intimately connected to an object known as the group ring. The structure of group rings is fiendishly complicated, but there is one type of group, a so-called torsion-free group, for which the structure is supposed to be simple.On the other hand, topology aims to understand the broad-brush structure of geometric objects, and algebraic topology applies tools from algebra to this problem. Many conjectures in algebraic topology concern the group ring, and can be related to purely algebraic properties of it.From around the 1950s, Irving Kaplansky set out a variety of conjectures in ring theory, and three of these - the idempotent conjecture, the zero-divisor conjecture and the unit conjecture - concern the algebraic structure of group rings of torsion-free groups. There is an intricate web of interdependencies between these conjectures and those of algebraic topology, and it leads to a conjecturally very elegant theory for these groups.However, it seems possible that the theory is not nearly so elegant. The unit conjecture in particular does not have direct topological equivalents. It has also been proved for far fewer classes of groups. Could it be false? Could the other two conjectures also be false?This project will study the three conjectures, looking for counterexamples rather than to prove the result for a class of groups. It aims to prove indeed that these groups are more complicated than previously thought, and perhaps the theory needs to be reworked.

群是对象对称性概念的数学体现，表示论研究这些对称性如何作用于空间。群的表示与称为群环的对象密切相关。群环的结构极其复杂，但有一种群，即所谓的无挠群，其结构应该是简单的。另一方面，拓扑旨在理解几何对象的粗略结构，而代数拓扑将代数的工具应用于这个问题。代数拓扑中的许多猜想都涉及群环，并且可以与群环的纯代数性质相关。从20世纪50年代左右开始，欧文·卡普兰斯基提出了环理论中的多种猜想，其中幂等猜想、零约数猜想和单位猜想等三个猜想涉及无挠群的群环的代数结构。这些猜想和代数拓扑猜想之间存在着一个错综复杂的相互依存关系网络，并且它导致了这些群的猜想上非常优雅的理论。然而，该理论似乎可能并不那么优雅。特别是单位猜想没有直接的拓扑等价物。这也已在少得多的群体类别中得到证明。会不会是假的？其他两个猜想是否也是错误的？本项目将研究这三个猜想，寻找反例而不是证明一类群的结果。它的目的确实是证明这些群体比以前想象的更复杂，也许这个理论需要重新设计。