Jakub Krásenský, Czech Technical University in Prague
"Criterion sets for quadratic forms over number fields"
By the celebrated 15 theorem of Conway and Schneeberger, a classical positive definite quadratic form over Z is universal if it represents each element of {1,2,3,5,6,7,10,14,15}. Moreover, this is the minimal set with this property. In 2005, B.M. Kim, M.-H. Kim and B.-K. Oh showed that such a finite criterion set exists in a much general setting, but the uniqueness of the criterion set is lost. Since then, the question of uniqueness for particular situations has been studied by several authors.
We will discuss the analogous questions for totally positive definite quadratic forms over totally real number fields. Here again, the existence of criterion sets for universality is known, and Lee determined the set for Q(sqrt5). We will show the uniqueness and a strong connection with indecomposable integers. A part of our uniqueness result is (to our best knowledge) new even over Z. This is joint work with G. Romeo and V. Kala.
Zoom link: https://uwaterloo.zoom.us/j/98937322498?pwd=a3RpZUhxTkd6LzFXTmcwdTBCMWs0QT09