Lecture I: Quadratic rings and Gauss composition (Manjul Bhargava)
We define the notion of "ring of rank n", and show how to classify rings
of ranks 1 and 2. We discuss ideal classes in "oriented
quadratic rings"
(i.e., rings of rank 2), and explain how they lead to Gauss's composition
law on binary quadratic forms.
Lecture II: Prehomogeneous vector spaces and 2 x 2 x 2 cubes (Wei
Ho)
We introduce the notion of a "prehomogeneous vector space", and explain
its connection to Gauss composition. We give a description of the
fundamental prehomogeneous vector space for quadratic fields -- namely,
the space of 2 x 2 x 2 cubes -- and sketch its many consequences,
following [1], leading to various "higher analogues" of Gauss composition.
[1] Bhargava, M. Higher composition laws I: A new view on Gauss
composition, and quadratic generalizations, Ann. of Math. 159 (2004),
no. 1, 217--250.
Lecture III: Gauss composition over an arbitrary commutative ring (Hendrik Lenstra)
We describe Kneser's approach to composition of binary quadratic forms
over an arbitrary commutative ring, via the use of Clifford algebras.
[2] M. Kneser: Composition of binary quadratic forms.
Journal of number theory 15 (1982) 406-413