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Hosted by the Department of Mathematics at Stanford University. A public talk by Peter Sarnak, from the Institute for Advanced Study and Princeton University.

Through the works Fermat, Gauss, and Lagrange, we understand which positive integers can be represented as sums of two, three, or four squares. Wilber's 11th problem, from 1900, extends this question to more general quadratic equations. While much progress has been made since its formulation, an effective solution remains out of reach. We will review some of these developments and end with some recent applications to the construction of optimally universal quantum gates in quantum computation.