Research Papers:

  • Hodge theory meets the minimal model program: a survey of log canonical and Du Bois singularities,   With Sándor Kovács. This paper surveys recent results on log canonical and Du Bois singularities, several new proofs are and some minor new results are also included. To appear in the Proceedings of the MSRI Workshop on the Topology of Stratified Spaces.
  • A refinement of strongly F-regular and sharply F-pure pairs. I point out that the current definitions of strong F-regularity and sharp F-purity do not behave as well as they should with respect to localization. I also point out how to fix this and how this new definition can be integrated into the existing theory.
  • Test ideals in non-Q-Gorenstein rings. We prove that the big test ideal τb(R) is the sum of test ideals τ(R, Δ) where Δ runs over divisors that make the pair (R, Δ) log-Q-Gorenstein.
  • Discreteness and rationality of F-jumping numbers on rings with singularities. With Manuel Blickle, Shunsuke Takagi and Wenliang Zhang. We prove discreteness and rationality of F-jumping numbers of pairs (R, a^t) when R is Q-Gorenstein with index not divisible by the characteristic p. To appear in Mathematische Annalen
  • On the number of compatibly Frobenius split subvarieties, prime F-ideals, and log canonical centers, with Kevin Tucker. We give a bound on the number of subvarieties compatibly Frobenius split with a fixed splitting of the Frobenius. To appear in Annales de L'Institut Fourier (Grenoble).
  • F-adjunction, we do a characteristic p > 0 analogue of inversion of adjunction along a center of log canonicity (at least in terms of relating singularities). Some applications are also explored. To appear in Algebra and Number Theory.
  • Globally F-regular and log Fano varieties, with Karen Smith. To appear in Advances in Mathematics.
  • Centers of F-purity, to appear in Mathematische Zeitschrift. We discuss a positive characteristic analogue of a notion from characteristic zero, centers of log canonicity (aka, log canonical centers). Click HERE for my slides on this topic from the conference in honor of Mel Hochster's 65th birthday..
  • The canonical sheaf of Du Bois singularities  With Sándor Kovács and Karen Smith. Submitted. You can also view Karen Smith giving a talk on these results HERE back in 2007.
  • Generalized test ideals, sharp F-purity, and sharp test elements    It has now appeared in Mathematical Research Letters.
  • Rational singularities associated to pairs    With Shunsuke Takagi. It has now appeared in the Michigan Mathematical Journal.
  • F-injective singularities are Du Bois    Based off of a part of my thesis. It has now appeared in The American Journal of Mathematics.
  • A simple characterization of Du Bois singularities    Also based on my thesis. It has now appeared in Composito Mathematica.
  • Gluing schemes and a scheme without closed points    This provides an explicit example that studying algebraic geometry can in fact be pointless (that joke was originally due to Paul Smith). I came up with this as a second year graduate student and managed to get it published. See
      Proceedings of the 2002 John H. Barrett Memorial Lectures Conference on Algebraic/Arithmetic Geometry, (eds Y. Kachi, S. Mulay, P. Tzermias). 2005.

    Research papers in preparation

  • A geometric algorithm for computing seminormalization, based off my thesis, in preparation. In principal, this algorithm could be implemented in Macaulay2 or a similar program (Macaulay2 needs a couple functions which have not yet been implemented). I described this algorithm in my Thesis, and if you CLICK HERE you can view the relevant portion of my thesis on the topic.
  • The behavior of test ideals and Frobenius splittings under finite maps, with Kevin Tucker.