What's Happening in Fine Hall

Gauss studied the classification of binary quadratic forms, i.e. expressions of the form ax^2+bxy+cy^2 with integer coefficients a,b,c, up to linear change of variables. The discriminant b^2-4ac is an invariant, and there are finitely many equivalence classes with a given invariant. Unexpectedly, the equivalence classes with a given discriminant form an abelian group. Cohen and Lenstra gave predictions for the structure of this finite abelian group for typical discriminants. I will explain these predictions, generalizations of which are a subject of active research, and sketch how they relate to natural questions in topology, algebraic geometry, and probability theory.