Title: Levels and Pythagoras numbers of commutative rings

Abstract:  The level s(R) of a commutative ring R is the smallest integer n such that -1 is a sum of n squares of elements in R.  Set s(R) = infinity if no such representation exists. The Pythagoras number p(R) is the smallest integer m such that every sum of squares of elements in R is already a sum of m squares in R.  Set p(R) = infinity if no such bound exists.  The study of levels and Pythagoras numbers of fields is a classical topic. Many results are known, but many open questions still remain.  The study of levels and Pythagoras numbers of arbitrary commutative rings is more recent.  I will survey known results and report on recent research with Detlev Hoffmann.