Prairie Mathematics Colloquium: Adam Clay

Similar to how the integers can be equipped with an ordering that is invariant with respect to addition, many groups can be equipped with an ordering that is invariant under the group operation. But aside from being a curious generalization of a standard algebraic structure, what role do orderable groups play in modern mathematics? In this talk, I will introduce orderable groups and answer this question by providing a brief overview of connections between orderable groups and areas of current research. The main focus, however, will be the L-space conjecture from low-dimensional topology, its connection with orderable groups, and how this conjecture has driven recent advancements in the field. In particular, I will explain how purely group-theoretic theorems have inspired topological results, and how topology might "give back" to group theory if the L-space conjecture turns out to be true.