Research | Reeve Garrett's personal u.osu.edu webpage

My area of research is commutative algebra. Commutative algebra generally concerns itself with the study of objects called commutative rings; that is, sets with addition and multiplication that satisfy their usual properties (like in the integers $\mathbb{Z}$ or polynomials in a single variable with fraction coefficients $\mathbb{Q}[X]$ ), e.g. both addition and multiplication are commutative ( $a+b=b+a$ ), 0 and 1 are in the set and have their usual properties (like in the integers), additive inverses (like negative numbers are additive inverses of their positive counterparts) exist, and the distributive property holds. Lots of interesting examples of commutative rings show up, including the rings that arise naturally from the study of varieties and schemes in the subject of algebraic geometry, the ring of entire functions on the complex plane (which is even a Prüfer domain!), rings of algebraic integers (arising in algebraic number theory), and the the ring of integer-valued polynomials $\hbox{Int}(\mathbb{Z})$ consisting of polynomials with rational (fraction) coefficients in one variable such that if an integer is substituted for the variable the result is always an integer. This last ring and rings like it are of special interest to me. As an undergraduate, under Professor David Rush’s direction, I studied rings of integer-valued polynomials, which include $\hbox{Int}(\mathbb{Z})$ as well as more general analogues where instead of the integers and the rational numbers we consider arbitrary integral domains $D$ and their fields of fractions $K$ , which we similarly denote $\hbox{Int} (D)$ , and rings of integer polynomials determined by subsets $E$ of $K$ , $\hbox{Int}(E,D)$ .

Another ring that is very easy to construct but is not well known to non-mathematicians is the collection $\mathbb{Z}_p$ of fractions such that the prime $p$ does not divide the denominator. Additionally, all the polynomials in one variable $X$ with coefficients that are fractions in $\mathbb{Z}_p$ form a ring $\mathbb{Z}_p[X]$ , and all the polynomials with coefficients in $\mathbb{Z}_p$ in $n$ variables ( $n$ any whole number), denoted $\mathbb{Z}_p$ [X_1, …, X_n], is too. If we allow our polynomials to have any fraction in its coefficients, we form a larger ring denoted $\mathbb{Q}$ [X_1, …, X_n]. Currently, I’m working on constructing rings with special hypotheses (namely, being integrally closed) between $\mathbb{Z}_p$ [X_1, …, X_n] and $\mathbb{Q}$ [X_1, …, X_n]. For more technical specifics about this problem, you can read my dissertation proposal (and written portion for my candidacy exam) that was written for mathematicians here.