My research is in commutative algebra, and is motivated by connections with birational algebraic geometry, the theory of differential operators, classical singularity theory, and convex geometry. I am especially interested in applications of the Frobenius endomorphism and related prime characteristic methods, local cohomology, and convexity to the study of algebraic varieties.
My research is partially supported by the National Science Foundation (NSF) through the grant DMS-1902321. In the recent past, my research was partially supported by the American Institute of Mathematics through an AIM SQuaRE award for collaborative research, the KU Office of Research through a KU Research GO Award, the KU New Faculty General Research Fund, and the NSF through the grant DMS-1600702, and through the Mathematical Sciences Postdoctoral Research Fellowship DMS-1304250.
I co-authored the following packages for Macaulay2 (M2), a software system devoted to supporting research in algebraic geometry and commutative algebra. These packages focus on explicit computation in prime characteristic, and include implementations of some of the algorithms developed in my research papers, but also a lot more.
- TestIdeals.m2
- M2 Documentation | GitHub
- FrobeniusThresholds.m2
- M2 Documentation | GitHub
- Bernstein-Sato functional equations, V-filtrations, and multiplier ideals of direct summands.
- Joint with Josep Àlvarez Montaner, Jack Jeffries, Luis Núñez Betancourt, Pedro Teixeira, and Emily E. Witt.
- Submitted.
- arXiv:1907.10017
- The FrobeniusThresholds package for Macaulay2.
- Joint with Karl Schwede, Pedro Teixeira, and Emily E. Witt.
- Submitted.
- arXiv:1906.09491
- Frobenius powers.
- Joint with Pedro Teixeira and Emily E. Witt.
- Math. Z. 296 (2020), no. 1-2 541-572
- arXiv:1802.02705
- The TestIdeals package for Macaulay2.
- Joint with multiple contributors.
- J. Softw. Algebra Geom. 9-2 (2019), 89-110.
- arXiv:1810.02770
- Frobenius powers of some monomial ideals.
- Joint with Pedro Teixeira and Emily E. Witt.
- J. Pure Appl. Algebra 224 (2020), no. 1, 66-85.
- arXiv:1808.09508
- Local Okounkov bodies and limits in prime characteristic.
- Joint with Jack Jeffries.
- Math. Ann. 372 (2018), no. 1-2, 139-178.
- arXiv:1701.02575
- Lyubeznik numbers and injective dimension of local cohomology in mixed characteristic.
- Joint with Luis Núñez Betancourt, Felipe Pérez, and Emily E. Witt.
- Trans. Amer. Math. Soc. 371 (2019), no. 11, 7533-7557.
- arXiv:1512.02298
- Cohomological dimension, Lyubeznik numbers, and connectedness in mixed characteristic.
- Joint with Luis Núñez Betancourt, Felipe Pérez, and Emily E. Witt.
- J. Algebra 514 (2018), 442-467.
- arXiv:1609.05846
- On the behavior of singularities at the F-pure threshold.
- Joint with Eric Canton, Karl Schwede, and Emily E. Witt.
- Illinois J. Math. 60 (2016), no. 3-4, 669-685.
- arXiv:1508.05427
- Local 𝔪-adic constancy of F-pure thresholds and test ideals.
- Joint with Luis Núñez Betancourt and Emily E. Witt.
- Math. Proc. Cambridge Philos. Soc. 164 (2018), no. 2, 285-295.
- arXiv:1801.05506
- F-threshold functions: Syzygy gap fractals and the two-variable homogeneous case.
- Joint with Pedro Teixeira.
- J. Symbolic Comput. 80 (2017), part 2, 451-483.
- arXiv:1404.5871
- F-pure thresholds of homogeneous polynomials.
- Joint with Luis Núñez Betancourt, Emily E. Witt, and Wenliang Zhang.
- Michigan Math. J. 65 (2016), no. 1, 57-87.
- arXiv:1404.3772
- F-purity versus log canonicity for polynomials.
- Nagoya Math. J. 224 (2016), no. 1, 10-36.
- arXiv:1112.2423
- F-invariants of diagonal hypersurfaces.
- Proc. Amer. Math. Soc. 143 (2015), no. 1, 87-104.
- arXiv:1112.2425
- F-pure thresholds of binomial hypersurfaces.
- Proc. Amer. Math. Soc. 142 (2014), no. 7, 2227-2242.
- arXiv:1112.2427
- F-purity of hypersurfaces.
- Math. Res. Lett. 19 (2012), no. 2, 389-401.
- arXiv:1112.2424
- Log canonical thresholds, F-pure thresholds, and non-standard extensions.
- Joint with Bhargav Bhatt, Lance Miller, and Mircea Mustaţă.
- Algebra Number Theory 6 (2012), no. 7, 1459-1482.
- arXiv:1106.0207