Research
Research Interests
My primary interests are in commutative and homological algebra, especially monomial ideals, Betti numbers, and combinatorial and topological methods in algebra. I am also interested in noncommutative algebra, combinatorics on words, algebraic combinatorics, representation theory, and solvable lattice models.
Publication
Noah Ripke and Phillip Yoon. “Characteristic Independence of Betti Numbers of Monomial Ideals in Five Variables.” Journal of Pure and Applied Algebra 230 (2026), 108354.
Current and Recent Research
Monomial ideals and Betti numbers
During the 2026 NSF REU/RET in Mathematics at California State University, Chico, I worked under the supervision of Guillermo Alesandroni on the dependence and independence of Betti numbers of monomial ideals on the characteristic of the base field. With Noah Ripke, we proved that the multigraded, graded, and total Betti numbers of every monomial ideal in five variables are independent of the characteristic. Our approach uses reductions to squarefree monomial ideals, Hochster’s formula, and the topology of simplicial complexes on at most five vertices. I also developed Python code reducing 7,580 squarefree ideals to 210 cases up to variable relabeling and carried out computational verification in Macaulay2. This work appeared in the Journal of Pure and Applied Algebra. I am currently investigating the corresponding questions in seven and eight variables, including constructions producing characteristic dependence in eight variables and the unresolved seven-variable case.
Growth of algebras and combinatorics on words
In ongoing joint work with Be’eri Greenfeld, we study growth and ideal structure in associative and monomial algebras through combinatorics on words. We constructed a finitely generated monomial algebra of quadratic growth with no largest nilpotent ideal, giving a negative answer to a question of L’vov over an arbitrary field at the slowest possible quadratic growth threshold. Our construction uses an infinite word with carefully controlled subword complexity, together with arguments involving hereditary languages. We are developing these results into a joint manuscript. I presented part of this work at Hunter College’s student research colloquium in April 2026.
Solvable lattice models and quantum superalgebras
Through Polymath Jr., I participated in collaborative research on solvable lattice models related to quantum superalgebras under the mentorship of Ben Brubaker. Our project studied partition functions of colored and supercolored lattice models and the local recurrence relations arising from Yang–Baxter-type equations. I worked on rank-three examples, vanishing conditions related to Bruhat order, and explicit computations used to test and understand the proposed recurrences. The project combined algebraic combinatorics, representation theory, and statistical-mechanical models. I presented this work jointly with Joey (Yizhou) Chen, Daniel Kumm, and Ioana Milea at the 2026 Joint Mathematics Meetings.