Perfectoid rings

In this talk I will discuss my recent computation of the NTC groups of perfectoid rings which have a system of pth power roots of unity and thus the Kgroups of the p-completed affine lineR〈x〉 over these rings relative to the ideal (x). This includes all perfect fields of positive characteristic, for which these groups vanish in non-negative degrees. This class of rings also contains many mixed characteristic rings, and perhaps surprisingly while the even nonnegative groups will still vanish, the odd groups will not.