In this talk, aimed at a general mathematical audience, I will revisit the classical problem of simplifying polynomials in one variable by Tschirnhaus transformations. Surprisingly, many of the old questions are still open. I will restate them in geometric terms and discuss recent progress in this area.

The classic Plateau's problem asks if given a smooth simple closed curve in space, does there exist a smooth orientable (embedded) surface-with-boundary spanning that curve, and having least area among all surfaces spanning the given curve? While the answer is in the affirmative, to solve Plateau's problem requires developing the theory of currents; these are generalized orientable surfaces-with-boundary. In this talk we study Plateau's problem for $C^{1,alpha}$ tangentially immersed boundaries.