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A snowball is a self-similar surface that is obtained in a fashion analogous to the snowflake curve. The purpose of this talk is to show that these spaces may serve as deterministic toy models for the Brownian map. This is a random metric space that appears as the scaling limit of triangulations of the $2$-sphere. The self-similarity of a snowball may be represented by a rational map. Snowballs are quasispheres, i.e., quasisymmetrically equivalent to the standard $2$-sphere. This is closely related to the visual metric associated to the rational map being quasisymmetrically equivalent to the spherical metric on the Riemann sphere.

COMD Stony Brook + MSRI Seminar Series: Snowballs, Quasispheres, And Rational Maps