This course is a continuation of the course that I gave in the Fall of 2021. Roughly, it will have four parts.
The first part will be an overview of the material from the Fall 2021. This will take about four first lectures.
In the second part, we will study in depth quantum sl_2, its representations, the braiding for quantum sl_2 and the specialization at roots of unity. After studying the category of finite dimensional representations for quantum sl_2 with divided powers, and the category of modules over small quantum sl_2 we will focus on the corresponding example of a modular tensor category and on its properties.
In the third part I will explain how to construct invariants of 3-manifolds and of 3-cobordisms using modular tensor categories and surgery of 3-manifolds, and I will discuss the general philosophy of topological quantum field theory (TQFT). After this we will focus on how to construct invariants of 3-manifolds and corresponding TQFT using triangulations of 3-manifolds. After this we will see how these two TQFT’s are related.
At the end of the course, in the fourth part, as much as time permits I will explain how these TQFT’s constructed combinatorially are related to a quantization of classical Chern-Simons topological field theory and will focus on various conjectures that follow from such (mathematically still conjectural relation).