Course Objectives: After a brief discussion of some of the historical aspects of the subject, we will begin with the modern perspective of a quadratic space introduced by Witt. Rather than work with individual forms, consider all forms over a given field $F$, and determine how to decompose them generically, but uniquely. This will lead to the introduction of the Witt ring which plays the role for quadratic forms that the Brauer group does for central simple algebras.
Following the study of quadratic spaces over general fields, we turn to valuation theory, local and global fields, and the introduction of the Hasse principle via the Hasse-Minkowski theorem.
Time permitting, forays into the integral theory and analytic applications. The local-global correspondence over rings is much more nuanced necessitating notions of the genus and spinor genus. Applications to modular forms.