I will discuss Garside's representation of elements of the braid group in
terms of "half- twists" and the corresponding solution to the Conjugacy Problem,
originally posed by Artin. If time permits, I will discuss some geometric
implications of this result.
I will discuss the proof that the exponential algebraic closure operator on
the
complex exponential field is isomorphic to the pregeometry which controls the
"pseudoexponential" field.