Abstract
Three different approaches lead recently to the solution of a longstanding open problem in algebra, namely to prove that the u-invariant of the function field of a curve over a p-adic number field is equal to eight. The u-invariant of a nonreal field is the smallest integer n such that every quadratic form in more than n varaibles over the field has a nontrivial zero. The different approaches are: (i) a combinatorial approach giving a far stronger result on systems of quadratic forms even over finitely generated extensions over any local number field; (ii) an approach using Galois cohomology and the construction of elements therein with given ramification; (iii) an approach called 'Field Patching' leading to new local-global principles for isotropy of quadratic forms over function fields of curves over a complete discrete valued field. Any of the three aproaches leads to more general results that so far are not attainable by the other two methods, for example (ii) and (iii) do currently not apply when p=2.
The proposed doctoral research project strives for a comparative analysis of the three methods and a better understanding of their strengths and limitations. As an application it is expected that the u-invariant as well as related field invariants for other fields can be determined in this manner.
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