Orderable groups and 3-manifolds
Abstract: This course will begin with an introduction to left-orderable and circularly orderable groups, and will focus on establishing the tools needed to tackle the problem of left-ordering fundamental groups of 3-manifolds–such as the Burns-Hale theorem, actions on R and S^1, and related Euler class arguments. With these tools in hand, we’ll move on to investigating some of the exceptional behaviour of left-orderability with respect to fundamental groups of 3-manifolds. Our chosen examples will draw inspiration from the L-space conjecture, with a focus on manifolds arising from Dehn surgery, and on Seifert fibred spaces. We’ll also encounter several open problems along the way, some completely algebraic having only to do with orderable groups, and others that are linked to special cases of the L-space conjecture.
TA: Junyu Lu
An introduction to Heegaard Floer homology
Abstract: We will give a broad overview of Heegaard Floer homology, focusing on applications to various topological questions. We will introduce the general setup and properties of the Heegaard Floer and knot Floer homology groups, discuss the surgery formula, and talk about L-spaces and L-space knots. If time permits, we will discuss some applications of Heegaard Floer homology to a variety of other questions, such as distinguishing exotic structures, sliceness obstructions, and so on.
TA: Hugo Zhou
An introduction to taut foliations in 3-manifolds
Abstract: This minicourse will be an introduction to taut foliations. We will introduce taut codimension-one foliations and their usefulness in understanding 3-manifolds. Topics will include the work of Gabai and Thurston relating minimal genus of surfaces to foliations, an assortment of constructions of taut foliations, the work of Eliashberg and Thurston relating taut foliations and contact structures, and the work of Ozsváth and Szabó on the contact class associated to a taut foliation.
TA: Atzimba Martinez