My research is in positive characteristic commutative algebra. In particular, I study the singularities of hypersurfaces (and more general rings) in characteristic *p*. Much of my work is motivated by the connection between between so-called *F-singularities* (the "*F*" here refers to the Frobenius endomorphism) and the classes of singularities arising in * birational algebraic geometry* over the complex numbers, typically defined via resolution of singularities.

## Articles

- Log canonical thresholds,
*F*-pure thresholds, and non-standard extensions (with Bhargav Bhatt, Lance Miller, and Mircea Mustaţă);*Algebra & Number Theory*, Vol. 6 (2012), No. 7, 1459-1482.We present a new relation between an invariant of singularities in characteristic zero (the

*log canonical threshold*) and an invariant of singularities defined via the Frobenius morphism in positive characteristic (the*F-pure threshold*). Our methods rely on combining results of Hara and Yoshida with non-standard constructions. -
*F*-pure thresholds of binomial hypersurfaces; to appear in*Proceedings of the AMS*.This article describes an algorithm for computing the

*F*-pure threshold of a binomial hypersurface; this algorithm remains valid in arbitrary characteristic, and generalizes earlier computations by Takafumi Shibuta and Shunsuke Takagi. The algorithm described in this article has recently been implemented in a*Macaulay2*package by Sara Malec, Karl Schwede, and Emily Witt. -
*F*-purity of hypersurfaces;*Mathematical Research Letters*, 19(02):1-13, 2012.We study

*F*-purity of pairs, and show (as is the case with log canonicity) that*F*-purity is preserved at the*F*-pure threshold. We also characterize when*F*-purity is equivalent to sharp*F*-purity, an alternate notion of purity for pairs introduced by Karl Schwede. We conclude by extending results describing the set of all*F*-pure thresholds to the most general setting. -
*F*-purity versus log canonicity for polynomials.This article considers the long-conjectured relationship between

*F*-purity and log canonicity for polynomials over the complex numbers. We define a non-degeneracy condition under which log canonicity and dense*F*-pure type are equivalent, and we also show that log canonicity corresponds, after reduction to characteristic*p*, to*F*-purity for very general polynomials. Note: This result has recently been generalized by Shunsuke Takagi. -
*F*-invariants of diagonal hypersurfaces; to appear in*Proceedings of the AMS*.We describe the higher jumping numbers and

*generalized test ideals*associated to diagonal hypersurfaces. Though these invariants are understood asymptotically (that is, as the characteristic tends to infinity), the results in this article allow one to produce examples in which the behavior in characteristic zero differs drastically from that in some fixed prime characteristic. -
*F*-pure thresholds of homogeneous polynomials (with Luis Núñez Betancourt, Emily E. Witt, and Wenliang Zhang).We give a description of the

*F*-pure threshold associated to a homogeneous (under an arbitrary grading) polynomial with an isolated singularity at the origin. Our methods allow us to produce seemingly "minimal lists" for these invariants, and also allows us to answer a question of Karl Schwede and myself regarding certain arithmetic properties of these invariants when they differ from their expected values predicted by characteristic zero considerations.