Constructing Counterexamples in Group Rings and Algebraic Topology

群是物体对称概念的数学体现，表征理论研究这些对称如何作用于空间。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，即所谓的无扭群，其结构被认为是简单的。另一方面，拓扑学旨在理解几何对象的大致结构，代数拓扑学将代数中的工具应用于此问题。代数拓扑中的许多猜想都与群环有关，并与群环的纯代数性质有关。大约从20世纪50年代开始，Irving Kaplansky在环理论中提出了各种各样的猜想，其中三个猜想——幂等猜想、零因子猜想和单位猜想——涉及无扭群群环的代数结构。在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赖的网络，这导致了这些群体在猜想上非常优雅的理论。然而，这个理论似乎并没有那么优雅。特别是单位猜想没有直接的拓扑等价。它也被证明适用于更少的群体。它会是假的吗？其他两个猜想也可能是错的吗？这个项目将研究这三个猜想，寻找反例，而不是证明一类群体的结果。它的目的是证明这些群体确实比以前认为的更复杂，也许这个理论需要重新设计。

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.