The Bergman kernel in complex analysis has always appeared to be an abstractly defined thing having to do with infinite sums of orthogonal functions, or with solutions to a PDE, or with extremal problems, or with the Riesz Representation Theorem. I will explain a way of looking at the kernel that connects it to algebra, and I will show how it can be used to convert problems in conformal mapping to problems in algebraic geometry. Would you believe that the Bergman kernel $K(z,w)$ of a multiply connected domain in the plane is just a rational combination of two explicit and simple analytic functions of one complex variable?