The Lie representation, a representation of dimension (n-1)! for the symmetric group on n letters, occurs within many contexts.
The purpose of this expository lecture is to describe some connections, concrete computations, as well as open problems concerning this
representation. Their common connection is via Dehn twists of Riemann surfaces together with their homological implications. Some topics
will include
(1) the cohomology ring of pure braid groups,
(2) the structure of homotopy string links and their invariants as developed by Milnor and Habegger-Lin,
(3) the infinitesimal braid relations as occurring in Vassiliev invariants of pure braids,
(4) complexity of algorithms for factoring complex polynomials, and
(5) certain groups of natural transformations.