Numerical semigroups have a surprisingly parallel theory to that of one-dimensional Cohen-Macaulay local rings, and can be used as a testing ground for results in commutative ring theory. They are also valuable examples in the factorization theory of monoids. My favorite results are those that involve a mix of ideas from commutative ring theory and the factorization theory of monoids.
Lance Bryant, James Hamblin, 2019, The Graphic Structure of a Numerical Semigroup, Rocky Mountain Journal of Mathematics, in Press.
Lance Bryant, James Hamblin, 2014, Position Vectors of Numerical Semigroups, Semigroup Forum, vol. 91, no. 1, pp. 28-38
Lance Bryant, James Hamblin, 2012, The Maximal Denumerant of a Numerical Semigroup, Semigroup Forum, vol. 86, no. 3, pp. 571-582
Lance Bryant, James Hamblin, Lenny Jones, 2012, Maximal Denumerant of a Numerical Semigroup with Embedding Dimension Less Than Four, Journal of Commutative Algebra, vol. 4, no. 4, pp. 489-503
Lance Bryant, James Hamblin, Lenny Jones, 2012, A Variation on the Money-Changing Problem, The American Mathematical Monthly, vol. 119, no. 5, p. 406
Lance Bryant, 2010, Goto Numbers of a Numerical Semigroup Ring and the Gorensteiness of Associated Graded Rings, Communications in Algebra, vol. 38, no. 6, pp. 2092-2128
Lance Bryant, Sarah Bryant, and Diana White, 2019, Striking the Right Chord: Math Circles Promote (Joyous) Professional Growth, “A Celebration of the EDGE Program’s Impact on the Mathematics Community and Beyond,” Springer, in Press.
Richard De Veaux & PCMI UFP Participants, 2017, Curriculum Guidelines for Undergraduate Programs in Data Science, Annual Review of Statistics and Its Application 4:1, 15-30
Erica Flapan & PCMI UFP Participants, 2016, Knots, Molecules, and the Universe: An Introduction to Topology, United States of America: American Mathematical Society