Description and aim of the Master Class 2007-2008

Quantum groups, affine Lie algebras and their applications

In the past decades powerful new tools from mathematical physics have emerged to distinguish low-dimensional topological manifolds. The resulting cross-fertilization between quantum field theory, topology and differential geometry, highlighted by the Fields Medal awards for Drinfeld, Jones and Witten in 1990, has led to a beautiful and rich new research area to which this Master Class provides a basic introduction.

Despite their different origins, quantum groups and affine Lie algebras are profoundly connected through their pivotal role in constructing quantum field theories. It is the Yang-Baxter equation from mathematical physics which has led to the development of quantum groups and to their applications in constructing topological invariants of 3-manifolds and knots. On the side of affine Lie algebras, the route goes through the differential geometric analysis of the Khniznik-Zamolodchikov equation.

This Master Class provides an introduction to quantum groups, semisimple and affine Lie algebras and quantum field theories, leading up to the construction of concrete and powerful topological invariants for 3-manifolds and knots. Besides the theoretical development, special attention is paid to the algorithmic and computational aspects of topological invariants using computer algebra.

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