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The Huneke-Wiegand conjecture is a decades-long open question in commutative algebra. Garc\'ia-S\'anchez and Leamer showed that a special case of this conjecture concerning numerical semigroup rings $\Bbbk[\Gamma]$ can be answered in the affirmative by locating certain arithmetic sequences within the numerical semigroup $\Gamma$. In this paper, we use their approach to prove the Huneke-Wiegand conjecture in the case where $\Gamma$ is generated by a generalized arithmetic sequence and showcase how visualizations can be leveraged to find the requisite arithmetic sequences.

The Park City Math Institute (PCMI) 2016 Summer Undergraduate Faculty Program met for the purpose of composing guidelines for undergraduate programs in Data Science. The group consisted of 25 undergraduate faculty from a variety of institutions in the U.S., primarily from the disciplines of mathematics, statistics and computer science. These guidelines are meant to provide some structure for institutions planning for or revising a major in Data Science.

A numerical monoid is an additive submonoid of the non-negative integers. Given a numerical monoid $S$, consider the family of "shifted" monoids $M_n$ obtained by adding $n$ to each generator of $S$. In this paper, we characterize the Ap\'ery set of $M_n$ in terms of the Ap\'ery set of the base monoid $S$ when $n$ is sufficiently large. We give a highly efficient algorithm for computing the Ap\'ery set of $M_n$ in this case, and prove that several numerical monoid invariants, such as the genus and Frobenius number, are eventually quasipolynomial as a function of $n$.

The catenary degree is an invariant that measures the distance between factorizations of elements within an atomic monoid. In this paper, we classify which finite subsets of $\mathbb Z_{\ge 0}$ occur as the set of catenary degrees of a numerical monoid (i.e., a co-finite, additive submonoid of $\mathbb Z_{\ge 0}$). In particular, we show that, with one exception, every finite subset of $\mathbb Z_{\ge 0}$ that can possibly occur as the set of catenary degrees of some atomic monoid is actually achieved by a numerical monoid.

A numerical monoid is an additive submonoid of the non-negative integers. Given a numerical monoid $S$, consider the family of "shifted" monoids $M_n$ obtained by adding $n$ to each generator of $S$. In this paper, we examine minimal relations among the generators of $M_n$ when $n$ is sufficiently large, culminating in a description that is periodic in the shift parameter $n$. We explore several applications to computation, combinatorial commutative algebra, and factorization theory., Comment: 15 pages, 2 figures

Let $M$ be a commutative cancellative monoid. The set $\Delta(M)$, which consists of all positive integers which are distances between consecutive factorization lengths of elements in $M$, is a widely studied object in the theory of nonunique factorizations. If $M$ is a Krull monoid with cyclic class group of order $n \ge 3$, then it is well-known that $\Delta(M) \subseteq \{1, \dots, n-2\}$. Moreover, equality holds for this containment when each class contains a prime divisor from $M$. In this note, we consider the question of determining which subsets of $\{1, \dots, n-2\}$ occur as the delta set of an individual element from $M$. We first prove for $x \in M$ that if $n - 2 \in \Delta(x)$, then $\Delta(x) = \{n-2\}$ (i.e., not all subsets of $\{1,\dots, n-2\}$ can be realized as delta sets of individual elements). We close by proving an Archimedean-type property for delta sets from Krull monoids with finite cyclic class group: for every natural number m, there exist a Krull monoid $M$ with finite cyclic class group such that $M$ has an element $x$ with $|\Delta(x)| \ge m$., Comment: 10 pages

Nonunique factorization in commutative monoids is often studied using factorization invariants, which assign to each monoid element a quantity determined by the factorization structure. For numerical monoids (co-finite, additive submonoids of the natural numbers), several factorization invariants have received much attention in the recent literature. In this survey article, we give an overview of the length set, elasticity, delta set, $\omega$-primality, and catenary degree invariants in the setting of numerical monoids. For each invariant, we present current major results in the literature and identify the primary open questions that remain.

Studying the factorization theory of numerical monoids relies on understanding several important factorization invariants, including length sets, delta sets, and $\omega$-primality. While progress in this field has been accelerated by the use of computer algebra systems, many existing algorithms are computationally infeasible for numerical monoids with several irreducible elements. In this paper, we present dynamic algorithms for the factorization set, length set, delta set, and $\omega$-primality in numerical monoids and demonstrate that these algorithms give significant improvements in runtime and memory usage. In describing our dynamic approach to computing $\omega$-primality, we extend the usual definition of this invariant to the quotient group of the monoid and show that several useful results naturally extend to this broader setting.

In an atomic, cancellative, commutative monoid $S$, the elasticity of an element provides a coarse measure of its non-unique factorizations by comparing the largest and smallest values in its set of factorization lengths (called its length set). In this paper, we show that the set of length sets $\mathcal L(S)$ for any arithmetical numerical monoid $S$ can be completely recovered from its set of elasticities $R(S)$; therefore, $R(S)$ is as strong a factorization invariant as $\mathcal L(S)$ in this setting. For general numerical monoids, we describe the set of elasticities as a specific collection of monotone increasing sequences with a common limit point of $\max R(S)$.

In commutative monoids, the $\omega$-value measures how far an element is from being prime. This invariant, which is important in understanding the factorization theory of monoids, has been the focus of much recent study. This paper provides detailed examples and an overview of known results on $\omega$-primality, including several recent and surprising contributions in the setting of numerical monoids. As many questions related to $\omega$-primality remain, we provide a list of open problems accessible to advanced undergraduate students and beginning graduate students.

In an atomic, cancellative, commutative monoid, the \omega-value measures how far an element is from being prime. In numerical monoids, we show that this invariant exhibits eventual quasilinearity (i.e., periodic linearity). We apply this result to describe the asymptotic behavior of the \omega-function for a general numerical monoid and give an explicit formula when the monoid has embedding dimension 2.

The Huneke-Wiegand conjecture has prompted much recent research in Commutative Algebra. In studying this conjecture for certain classes of rings, Garc\'ia-S\'anchez and Leamer construct a monoid S_\Gamma^s whose elements correspond to arithmetic sequences in a numerical monoid \Gamma of step size s. These monoids, which we call Leamer monoids, possess a very interesting factorization theory that is significantly different from the numerical monoids from which they are derived. In this paper, we offer much of the foundational theory of Leamer monoids, including an analysis of their atomic structure, and investigate certain factorization invariants. Furthermore, when S_\Gamma^s is an arithmetical Leamer monoid, we give an exact description of its atoms and use this to provide explicit formulae for its Delta set and catenary degree.

We show that the Kakimizu complex of minimal genus Seifert surfaces for a knot in the 3-sphere is quasi-isometric to a Euclidean integer lattice $\mathbb Z^n$ for some $n \geq 0$., Comment: 12 pages. Improvements to the exposition made in version 2