Learning seminar in geometry and topology
Rings and Modules seminar

What I do, non-technical

I work in an area of mathematics that deals with abstractions of a common thing that we see in everyday life--specifically, the numbers that we all use every day. The abstract generalization of numbers is something that is known as a group, and my specialization is group theory. One explanation of groups that you might see elsewhere is that they arose historically from the study of symmetries of geometric figures, rather than as an abstraction of our number system, and this is completely true. However this historical explanation illustrates a sort of group that I do not study, because this perspective on groups neglects one of the important properties that I like to have in every group that I think about: An order on the elements. By this, I mean that the numbers we use in day-to-day life are ordered (in the sense that 1 is smaller than 2, and 2 is smaller than 3, and so on...) and I usually study groups that have a similar structure--known as ordered groups.

For math students

My specialization lies in group theory, and specifically ordered groups. While my primary expertise are algebraic, ordered groups are part of a rich theory that is connected to many other parts of mathematics, and it is these connections that are my main focus. For example, there are connections between topology and ordered groups that arise in several ways. For example, one can make a topological space out of the collection of all orders on a group, and use the topology of that space to make exciting algebraic conclusions. Or, if you venture into the realm of algebraic topology, then the fundamental group of a space may or may not be an ordered group, and the study of orderability of these groups has become a very subtle and exciting question.

For mathematicians

I have focused for several years on orderability and circular orderability of groups, and their connections to topology. My interests are both in developing new theorems and results in the field of ordered groups, and in applying these new results to other areas of mathematics. As a first area of application and interest, the L-space conjecture from low-dimensional topology has been a point of focus of my work for some time. My research deals with the left-orderability aspect of this conjecture, and specifically on ordering fundamental groups of manifolds that arise from topological operations such as Dehn filling or gluing together two 3-manifolds along a common toral boundary component. I have also spent some time developing new criteria for left-orderability that may be of some use in pursuing this conjecture. I have also recently taken an interest in a particular aspect of descriptive set theory: the study of the countable Borel equivalence relations that arise from considering the Polish space of all left-orderings of a countable group G equipped with the natural G-action. It turns out that these sorts of equivalence relations can encode varying levels of Borel complexity, and that there are questions in descriptive set theory which admit tantalizing parallels in the field of ordered groups--perhaps hinting at a deeper connection than we currently know.

Links to other mathematicians

Tyrone Ghaswala
Steve Boyer
Tali Pinsky
Filippo Calderoni
Cristóbal Rivas
Kathryn Mann
Jason Bell
Dale Rolfsen
Patrick Naylor