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Title:
Weil-Petersson curves, knot energies, traveling salesman theorems, and minimal surfaces
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Abstract:
Weil-Petersson curves are a class of rectifiable closed Jordan 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 was motivated by string theory and makes the space of closed loops into a Hilbert manifold, but the same class of curves also arises naturally in complex analysis, geometric measure theory, probability theory, knot theory, computer vision, and other areas. No geometric description of Weil-Petersson curves was known until 2019, but there are now more than twenty equivalent conditions, many of which extend to curves in higher dimensions and remain equivalent there. 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 arclength has finite renormalized electrostatic energy for an inverse cube law, and yet another says a curve if Weil-Petersson iff it bounds a minimal surface in hyperbolic 3-space that has finite total curvature, or equivalently, finite renormalized area. I will discuss these and several other characterizations and sketch why they are all equivalent to each other. The lecture will contain many pictures, several definitions, but not too many proofs or technical details.
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