Frobenius closure and prime characteristic singularities

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT REQUEST OF AUTHOR.] This dissertation outlines several results about prime characteristic singularities for which the nilpotent part under the induced Frobenius action on local cohomology is either finite colength or the entire module, collectively referred to here as nilpotent singularities. First, we establish a sufficient condition for the finiteness of the Frobenius test exponent for a local ring and apply it to conclude that nilpotent singularities have finite Frobenius test exponent. In joint work with Jennifer Kenkel, Thomas Polstra, and Austyn Simpson, we show that under mild conditions nilpotent singularities descend and ascend along faithfully flat maps. Consequently, we then prove that the loci of primes which are weakly F-nilpotent and F-nilpotent are open in the Zariski topology for rings which are either F-finite or essentially of fiiite type over an excellent local ring.