Research

Papers

• cl-prereductions, i-postexpansions, and related structures, with Janet Vassilev (2023). 

  • Arxiv Link 

  • Abstract: Expanding on the work of Kemp, Ratliff and Shah, for any closure cl defined on a class of modules over a Noetherian ring, we develop the theory of cl-prereductions of submodules. For any interior i on a class of R -modules, we also develop the theory of {\rm i}-postexpansions. Using the duality of Epstein, R.G. and Vassilev, we show that if i is the interior dual to cl , then these notions are in fact dual to each other. We consider the cl-precore (i-postcore), the intersection of all cl-prereductions i-postexpansions) of a submodule and the cl-prehull (i-posthull), the sum of all cl-prereductions (i-postexpansions) of a submodule and give comparisons with the cl-core (i-hull). We further give a classification of cl-prereductions of cl-closed ideals of a Noetherian ring where cl is a closure with a special part.

• Pair operations, with Janet Vassilev and Nicolas Berkopec (2023). 

  • Abstract: In progress.

Invited Talks

• •“The Duality of Pair Operations.” October 2023

  • AMS Central Sectional meeting, Creighton University, NE.

• "TBD" September 2023

  • Algebra Seminar, University of Kentucky, KY.

Other Talks and Poster Presentations

• “Constructing Pair Operations.” October 2023

  • Poster, AWM Research Symposium, Clark Atlanta University, GA.

• "Symmetry Samplers in the Fiber Arts" August 2023

  • Math and More, Coffee and Talks, University of New Mexico, NM.

• "What’s Up With Pair Operations?" August 2023

  • Algebra and Geometry Seminar, University of New Mexico, NM.

• “Prereductions, Postexpansions, and Related Structures.” April 2023

  • Lightning Talk, Commutative Algebra in the South, University of Alabama, AL.

• “Prereductions, Postexpansions, Prehulls, and Postcores.” March 2023

  • Poster, Southwest Local Algebra Meeting, University of North Texas, TX.

• “cl-Prereductions and i-Postexpansions.” October 2022

  • Algebra and Geometry Seminar, University of New Mexico, NM. 

Select Workshops Attended

• Pan-American School in Commutative Algebra (PASCA), 2022.

• Building Relationships for an Inclusive and Diverse Group of Emerging Students (BRIDGES), 2021.

• Undergraduate Summer School on Homological Algebra of Mirror Symmetry, 2019.