On translation invariant quadratic forms

Solutions to translation invariant linear forms in dense sets  (for example: k-term arithmetic progressions), have been studied extensively in additive combinatorics and number theory. Finding solutions to translation invariant quadratic forms is a natural generalization and at the same time a simple instance of the hard general problem of solving diophantine equations in unstructured sets. In this talk I will explain how to modify the  classical circle method approach to obtain quantitative results  for quadratic forms with at least 17 variables.