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Friday, February 1, 2019

**Abstract:** Given a planar domain with smooth boundary, one can associate its heat kernel, a time dependent operator whose trace admits an asymptotic expansion in t. The coefficients in this expansion turn out to all be geometric/topological invariants of the domain. However, by considering a smooth family of domains converging to a polygon, one can conclude that these heat trace coefficients are not continuous under such domain deformation. In this talk I’ll describe work of Mazzeo-Rowlett which recasts this apparent anomaly using renormalized invariants. I’ll also use it as an excuse to talk about uncommon but useful techniques in the study of linear PDEs e.g.: domain blow-ups, polyhomogeneous expansions, and more.