Abstracts for Quantum topology seminar

Classical quantum topology studies polynomial invariants of links arising from the representation of quantum groups — in particular, the sln-Jones polynomial, associated to quantum sln. Around 2000, it was realized that some of these polynomial invariants generalize (“categorify”) to homological invariants — in particular, Khovanov–Rozansky homology categorifying the sln-Jones polynomial. This homology can be defined in various ways, including using matrix factorisation, category O, the algebraic geometry of flag varieties, or defect TQFTs. String diagrams for higher categories, i.e.

This talk is about the homological algebra of a class of associative algebras which are often called "diagrammatic". I'll try to give an overview which is approachable regardless of whether you are in rep theory, homotopy theory, or knot theory. First I'll describe how people got interested in these questions: roughly speaking, they were interested in a phenomenon called homological stability, and wanted to know whether it can be done for algebras. I'll try to give an overview of how those results go for the most studied examples: Temperley--Lieb, Brauer, and partition algebras.

I will discuss foliated open book decompositions of 3 manifolds, bypass attachments, and how one can perform a bypass attachment to a foliated open book.