Description and aim of the Master Class 2007-2008
Quantum groups, affine Lie algebras and
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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