Research Description
My primary research concerns algebraic order theory, partially ordered sets, and commutative algebra.
Published and Submitted Works
2025 Chain Conditions and Optimal Elements in Generalized Union-Closed Families of sets. https://doi.org/10.48550/arXiv.2412.18740 . Accepted for publication. To appear in Order.
2024 (with A. Lee) Rigid-Flexible Values for Embeddings of Four-Dimensional Ellipsoids into Almost-Cubes. https://doi.org/10.48550/arXiv.2402.16223. Accepted for publication. To appear in Advances in Geometry.
2023 (with S. Loepp) Every Finite Poset is Isomorphic to a Saturated Subset of the Spectrum of a Noetherian UFD, Journal of Algebra, Volume 643, Pages 340-370, https://doi.org/10.1016/j.jalgebra.2023.11.042.
2023 A Structural Invariant On Certain Two-Dimensional Noetherian Partially Ordered Sets, Rocky Mountain Journal of Mathematics, Volume 53, No. 3, 711-724, DOI: 10.1216/rmj.2023.53.711
2022 (with S. Loepp) Gluing minimal prime ideals in local rings, Communications in Algebra, 51:1, 239-247, DOI: 10.1080/00927872.2022.2096226
2021 Enlarging localized polynomial rings while preserving their prime ideal structure, Journal of Algebra, Volume 575, Pages 14-30, https://doi.org/10.1016/j.jalgebra.2021.02.011.
2013 (with E. Burger, D. Clyde, G. Shin, and Z. Wang) Canonical Diophantine representations of natural numbers with respect to quadratic “bases.” J. Number Theory 133, no. 4, 1372-1388.
2012 (with E. Burger, D. Clyde, G. Shin, and Z. Wang) A generalization of a theorem of Lekkerkerker to Ostrowski’s decomposition of natural numbers. Acta Arith. 153 no. 3, 217-249.
Active Works
Union-closed families of sets stable under transitive group actions.
Understanding the structure of ideal lattices arising from commutative rings.
New and Emerging Projects
Degree functions on Noetherian orders.
Counting antichains in inductive sequences of finite partial orders.
Other Works
On Two Dimensional Semi-Local Noetherian Spectra. https://doi.org/10.48550/arXiv.1801.02758
Current and Past Research Students
Troy Larsen
Eric Gazin
Alex Fedor