Eric Canton

Pure and Applied Mathematics Quarterly, vol. 16 (2020) no. 5, article 5, pp.1465–1532

https://dx.doi.org/10.4310/PAMQ.2020.v16.n5.a5

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

https://doi.org/10.1016/j.jpaa.2016.01.006

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

https://doi.org/10.1215/ijm/1506067286

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.