CohenLenstra theory
CohenLenstra theory

Will Sawin, Princeton University
Fine Hall Common Room
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^24ac 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.