Math Colloquium: Statistics of quadratic congruences and generalizations

We present the uniform distribution of roots of quadratic congruences on the unit circle. Several proofs have been rediscovered over the years. I will review some of the very different methods involved -- ergodic theory, exponential sums, automorphic forms. It constitutes an excellent introduction to the more delicate Linnik problems. I will then proceed to describe new generalizations. These have far reaching applications to arithmetic geometry and independence of Heegner points on rational elliptic curves.