I enjoy extracting trends and studying dynamics of data. My background is in mathematical analysis of many kinds— I am a peer-reviewed author in algebraic geometry and commutative algebra, and am a core author of apps, scientific packages, and data pipelines at Arcascope.
My research in pure mathematics is in the intersection of algebraic geometry, commutative algebra, and harmonic analysis.
Pure and Applied Mathematics Quarterly, vol. 16 (2020) no. 5, article 5, pp.1465–1532
Based on PhD Thesis.
Harnessing the powerful theory of Berkovich spaces, I prove a collection of theorems about singular algebraic varieties over positive characteristic fields. Reflecting deep connections between Berkovich spaces and Fourier transforms, I convert difficult bounds on singularities (derivative condition) to linear inequalities on transforms (algebraic condition).
Journal of Pure and Applied Algebra, Vol. 220, Issue 8, August 2016, pp.2879–2885
By giving new, explicit formulae for the Frobenius action on local cohomology of certain kinds of projective varieties, I extend theorems of B. Bhatt and A. Singh relating singularities of these varieties to their cohomological structure. The key idea is to apply the local cohomology functors to graded free resolutions of the varieties' ideals.
Illinois Journal of Mathematics, Volume 60, Number 3-4 (2016), pp.669-685
Joint with Daniel Hernández, Karl Schwede, and Emily E. Witt.
We study new families of varieties where the log canonical and F-pure thresholds do not coincide; these are two numerical measures of singularities that have deep, mysterious connections. We also study a volume called the F-signature whose derivatives I related to other volumetric invariants of singularities as an undergraduate, working with Karl.
I spoke at Mathematisches Forschungsinstitut Oberwolfach on my thesis work from a functional-analytic perspective. Techniques linking splittings of the Frobenius morphism to minimal model program singularities take the form of theorems about operator norms. This is the technical report associated with my talk.