Research

My research uses tools from commutative algebra to study singularities in algebraic geometry, especially in positive and mixed characteristic, with applications to arithmetic and birational geometry.

The Frobenius map (\(x \mapsto x^p\)) is a central tool in positive characteristic geometry—by imposing requirements on the Frobenius map associated to a ring (or scheme), one can impose significant control on the ring (or scheme). This is useful, for instance, in controlling singularities arising in the Minimal Model Program and controlling arithmetic behavior like ordinarity. These Frobenius-defined invariants are called \(F\)-Singularities, and have been a central topic of study in commutative algebra and algebraic geometry for over 50 years. I study so-called quasi-\(F\)-singularities, weaker conditions which instead impose requirements on a \(p\)-nilpotent thickening of Frobenius, and higher-\(F\)-Singularities, stronger conditions defined by the action of Cartier operators (a generalization of Frobenius) on differential forms.

The field of mixed characteristic singularities by contrast is much newer and far more complex, borrowing tools from \(p\)-adic Hodge theory and Scholze’s theory of perfectoid spaces. Here I am interested in the plus-pure threshold, a new and mysterious asymptotic invariant that translates between the \(F\)-pure threshold in positive characteristic and the log canonical threshold over the complex numbers. I am currently working (with many others) on computing this invariant for a number of important examples. I am also interested in developing both higher and quasi analogues of singularities in mixed characteristic, and believe our plus-pure threshold computations will provide many examples with which to flesh out these ideas.