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Tuesday, April 2, 2019

**Abstract:** The Weil-Petersson metric is a Riemannian metric on the Teichmuller space which is natural in the sense that it comes from and reflects the geometry of the hyperbolic metrics on the underlying surface. Motivated by foundational work of McMullen, Pollicott-Sharp (and later Kao) proposed an analogous metric for the moduli space of metrics on a fixed graph. We study this metric and completely characterize its completion in the case of a rose. In this talk, we’ll introduce the Weil-Petersson metric and do our best to motivate the definitions so that no advanced prior knowledge of the subject will be necessary. This represents joint work with Matt Clay and Yo’av Rieck.