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The fifth edition of the ALGAR summer school is dedicated to sums of squares and systems of quadratic forms over arithmetic fields. In particular we will look at field invariants, such as the Pythagoras number and the u-invariant, for function fields over local and global fields. 

E. Artin conjectured that systems of r quadratic forms over p-adic number fields in more than 4r variables are isotropic. An extended variant of this conjecture (for forms of higher degree) has turned out to be false. As stated here the conjecture was recently proven by D.R. Heath-Brown and D. Leep. One goal is to present this result and its ingredients, in particular from invariant theory. 

The Pythagoras number of a ring is the smallest integer n such that any totally positive element is a sum of n squares. Determining its value is a very classical problem. It is conjectured that the Pythagoras number of a function field of a curve over a number field is at most 5. For rational function fields this was shown by Y. Pourchet in 1971. More generally the upper bound 6 was established by F. Pop in 1990. A second goal is to present these two results.