**Title:** Invariants of 3-manifolds and topological quantum field theory. The past, the present and the future.

**Speaker:** Nicolai Reshetikhin

**Speaker Info:** Tsinghua University

**Brief Description:**

**Special Note**:

**Abstract:**

The goal of this lecture is an overview of the concept of Topological Quantum Field Theory (TQFT). First I recall Atiyah's concept of TQFT as a functor from the category of cobordisms to the category of vector spaces and its generalization to manifolds with corners.Then we focus on the path integral framework of Chern-Simons TQFT and on possible ways to make it mathematically meaningful. Here we give a short summary of works of Witten, Axelrod and Singer, Kontsevich and others where the path integral is treated perturbatively (semiclassically) in terms of Feynman diagrams and will mention some open problems.

Quantum Chern-Simons theory is closely related to the Wess-Zumino-Witten conformal quantum field theory. One of the central parts of this theory is the Knizhnik-Zamolodchikov flat connection. Following Drinfeld's work we will see how to use this equation to construct a braided tensor category which is a deformation of the category of finite dimensional modules ${\mathfrak g}$-mod over a simple finite dimensional Lie algebra $\mathfrak g$ and how to use it to construct invariants of tangles.

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