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Speaker: Christopher Bishop, Stony Brook University
Title: Weil-Petersson curves, traveling salesman theorems, and minimal surfaces
Abstract: Weil-Petersson curves are a class of rectifiable closed curves in the plane, defined as the closure of the smooth curves with respect to the Weil-Petersson metric defined by Takhtajan and Teo in 2009. Their work solved a problem from string theory by making the space of closed loops into a Hilbert manifold, but the same class of curves also arises naturally in complex analysis, probability theory, knot theory, applied mathematics, and other areas. No geometric description of Weil-Petersson curves was known until 2019, but there are now more than thirty equivalent conditions. One involves inscribed polygons and can be explained to a calculus student. Another is a strengthening of Peter Jones's traveling salesman theorem characterizing rectifiable curves. A third says a curve is Weil-Petersson iff it bounds a minimal surface in hyperbolic space that has finite total curvature. I will discuss these and several other characterizations, and sketch why they are all equivalent to each other.