I am a commutative algebraist, and my research 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.
I am an active member of Macaulay2 (M2) community. M2 is a software system devoted to supporting research in algebraic geometry and commutative algebra, and I have co-authored the following packages, which focus on explicit computation in prime characteristic, and include implementations of some of the algorithms developed in my research, but also a lot more.
- TestIdeals.m2 | M2 Documentation, GitHub
- FrobeniusThresholds.m2 | M2 Documentation, GitHub
My research is partially supported by the National Science Foundation (NSF) through the grant DMS-1902321.
- Fractal programs, arithmetic programs, and the Frobenius powers of monomial ideals. Pedro Teixeira, and Emily E. Witt.
- In preparation
- The Bernstein-Sato polynomial of a simple algebroid curve via reduction to prime characteristic .
- With Emily E. Witt.
- In preparation
- Jumping numbers of F-pure submodules
- With Alessandro De Stefani, Luis Núñez Betancourt, and Emily E. Witt.
- Submitted
- Bernstein's inequality and holonomicity for certain singular rings.
- With Josep Àlvarez Montaner, Jack Jeffries, Luis Núñez Betancourt, Pedro Teixeira, and Emily E. Witt.
- Submitted (2021).
- arXiv:2103.02986
- Bernstein-Sato functional equations, V-filtrations, and multiplier ideals of direct summands.
- With Josep Àlvarez Montaner, Jack Jeffries, Luis Núñez Betancourt, Pedro Teixeira, and Emily E. Witt.
- To appear in Communications in Contemporary Mathematics.
- arXiv:1907.10017
- The FrobeniusThresholds package for Macaulay2.
- With Karl Schwede, Pedro Teixeira, and Emily E. Witt.
- Journal of Software for Algebra and Geometry 11 (2021), no. 1, 25-39
- arXiv:1906.09491
- Frobenius powers.
- With Pedro Teixeira and Emily E. Witt.
- Math. Z. 296 (2020), no. 1-2 541-572
- arXiv:1802.02705
- The TestIdeals package for Macaulay2.
- With multiple contributors.
- J. Softw. Algebra Geom. 9-2 (2019), 89-110.
- arXiv:1810.02770
- Frobenius powers of some monomial ideals.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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.
- With Pedro Teixeira.
- J. Symbolic Comput. 80 (2017), part 2, 451-483.
- arXiv:1404.5871
- F-pure thresholds of homogeneous polynomials.
- 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.
- Journal version (updated)
- 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.
- With Bhargav Bhatt, Lance Miller, and Mircea Mustaţă.
- Algebra Number Theory 6 (2012), no. 7, 1459-1482.
- arXiv:1106.0207