This series of introductory talks will discuss some of the issues and pathologies one runs into when exploring birational geometry in positive characteristic. Some time will be spent talking about ideas coming from commutative algebra and their applications, including connections between singularities from the MMP with singularities arising in commutative algebra.

Suppose that R is a local ring of mixed characteristic. Using recent breakthrough results of André on the existence of big Cohen-Macaulay algebras, we defined a mixed characteristic analog of the multiplier ideal / test ideal and show it satisfies many of the same formal properties as its equal characteristic brethren. Using the same ideas, we show that if R is mixed characteristic and local and R/pR has F-rational or F-regular singularities, then R itself has analgous singularities in mixed characteristic. This is joint work with Linquan Ma