Ehsaan Hossain, Pure Mathematics, University of Waterloo
"The Invariant Basis Number Property"
In linear algebra, the "dimension'' of a vector space can be defined uniquely --- every vector space has a basis, and all bases have the same cardinality. But this is a special advantage of working over a field. What about modules over a general ring $R$? It's possible that an $R$-module does not possess a basis, and even if it does, there may be bases of different cardinalities. In fact there is an easy example of a module possessing a basis is of cardinality $n$, for all $n\in \mathbf{N}$. Can we find an example of an $R$-module with bases of cardinality 2 and 3, but not 5? How far can we push this? We will see a group, called $K_0(R)$, which encodes this information (at least partially). I also hope to convince you that this investigation leads inevitably to the invention of Leavitt path algebras.