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Title: Abstract: The angle defect -- 2 Pi minus the cone angle at a vertex -- is the commonly used discretization of the Gaussian curvature for piecewise flat surfaces. However, it does not possess one of the principal features of its smooth counterpart -- upon scaling the surface by a factor r, the smooth Gaussian curvature is scaled by the factor of 1/r^2 , whereas the angle defect is invariant under global scaling.

In my talk I will introduce a new discretization of the Gaussian curvature, that preserves the properties of the angle defect and in addition reflects the scaling behavior of the smooth Gaussian curvature. I will also answer the accompanying Uniformization question: Does every discrete conformal class of a piecewise flat surface contain a metric with constant discrete Gaussian curvature? And if so, is this metric unique?

The results I will present in this talk constitute a part of my PhD research, which was supervised by prof. Boris Springborn.