A geometric approach to Gauss composition with applications to proper surfaces in the 4-ball

In Disquisitiones Arithmeticae, motivated by the problem of characterizing which integers are representable as x^2+ny^2, Gauss described a composition law that turns primitive integral binary quadratic forms of a fixed discriminant D into a group. For this integral binary quadratic forms – expressions of the form ax^2+bxy+cy^2 – are considered up to linear coordinate changes in the variables x and y.

We discuss a new perspective on Gauss composition, which relates to Bhargava's celebrated cube law. Our perspective grew out of a problem in 3.5-dimensional topology: we tackle the problem of distinguishing Seifert surfaces in the 4-ball, and use our new perspective to provide robust constructions of pairs of Seifert surfaces that are not isotopic in the 4-ball. This puts recent examples of Hayden-Kim-Miller-Sundberg-Park concerning a question of Livingston into a more general framework.

Based on joint work with Menny Aka, Alison Beth Miller, and Andreas Wieser.