Given a quadratic form (homogeneous degree two polynomial) q over a field k, some basic questions one can ask are

- Does q have a non-trivial zero (is q
*isotropic*)? - Which non-zero elements of k are represented by q?
- Does q represent all non-zero elements of k (is q
*universal*)?

Over a global field F, the Hasse-Minkowski Theorem, which is one of the first examples of a

*local-global principle*, allows us to use answers to these questions over the completions of F to form answers to these questions over F itself. In this talk, we will focus primarily on quadratic forms over*semi-global fields*(function fields of curves over complete discretely valued fields), and see how a local-global principle of Harbater, Hartmann, and Krashen can be used to study universal quadratic forms over semi-global fields.