According to the late Stephen Hawking “The zeta function technique can be applied to calculate the partition functions for thermal gravitons and matter quanta on black hole and de Sitter backgrounds.”  The aforementioned zeta function technique is none other than the zeta regularization of the determinant of the Laplace operator.  Here we consider the determinant on singular surfaces with edges and conical singularities.  Whereas the determinant is a global quantity, its variation by a smooth conformal factor is local.  We shall consider variations which change the cone angles, corresponding to a singular conformal factor.  The first main result is an explicit formula expressing the dependence of the determinant on the cone angle.  In the second part of the talk, we move to one of the few examples for which one can actually compute the determinant in closed form:  rectangles.  The next main result is that amongst all rectangles of a given area, the square uniquely maximizes the determinant.  Interestingly, the proof combines ingredients from analytic number theory together with spectral theory, resulting in a somewhat unusual mathematical 火锅.

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