Constructing Counterexamples in Group Rings and Algebraic Topology

群是对象对称性概念的数学体现，而表象论研究这些对称性如何作用于空间。群的表示与一个称为群环的对象密切相关。群环的结构极其复杂，但有一类群，即所谓的无挠群，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粗刷结构，代数拓扑学将代数的工具应用于这一问题。在代数拓扑中的许多命题涉及群环，并且可以涉及到它的纯代数性质。从20世纪50年代左右，欧文卡普兰斯基提出了各种命题在环理论，其中三个-幂等猜想，零因子猜想和单位猜想-关注的代数结构群环的挠自由群。在这些拓扑和代数拓扑之间有一个错综复杂的相互依赖的网络，它导致了一个理论上非常优雅的理论，然而，它似乎可能不是那么优雅。特别是单位猜想没有直接的拓扑等价物。它也被证明为少得多的类的群体。会不会是假的？另外两个假设也是假的吗？这个项目将研究这三个命题，寻找反例，而不是证明一类群的结果。它的目的是证明这些群体确实比以前认为的更复杂，也许这个理论需要修改。

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.