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Algebra Seminar

Monday, March 27, 2023 - 3:30pm

Connor Cassady

University of Pennsylvania


University of Pennsylvania


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.