How do you conformally weld? That is, given a welding homeomorphism $\varphi:\mathbb{S}^1 \rightarrow \mathbb{S}^1$, how do you compute the associated conformal maps? The measurable Riemann mapping theorem provides a non-constructive answer for quasisymmetric weldings, but sometimes an explicit approximation is helpful. A beautiful algorithm developed by Don Marshall and Lennart Carleson in the early 1980's, called the zipper, provides a numerically robust approach by composing large numbers of "slit" mappings. We give the first convergence results for the zipper by shifting the problem from weldings to driving functions. In the process, we also give some results on minimal-energy curves and, if time permits, draw connections to universal Teichmuller space. There will be many pictures and the talk should be widely accessible.

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