**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:**

We first recall the definition of quantized universal enveloping algebras $U_q({\mathfrak g})$ and how they are related to the deformation of ${\mathfrak g}$-mod constructed in the first lecture. Then we construct invariants of tangled graphs corresponding to finite dimensional representations of $U_q({\mathfrak g})$.After a brief recollection of the surgery construction of 3-manifolds and Kirby calculus, we see that invariants of 3-manifolds can be regarded as invariants of links in $S^3$ which are invariant with respect to Kirby moves. We will construct such invariants using modular tensor categories (MTC).

Examples MTC are certain quotients of the category of finite dimensional modules over quantized universal enveloping algebras at a root unity. We will see how this works for $sl_2$ and will finish with some open problems, in particular on the semiclassical conjecture about the relation of these invariants to Feynman diagrams discussed in Lecture 1.

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