Given a field \(F\) adjoined with a polynomial variable \(x\), we can create a numerical semigroup algebra by restricting the allowed exponents to members of the semigroup \(S\). This is denoted \(F[S]\). Since \(F[x]\) is a unique factorization domain, we can explore the irreducible elements of the algebra \(F[S]\) by looking at their factorization in the full polynomial ring. In this talk we explore irreducible elements for the semigroup algebra using the field \(\mathbb F_2\) and the semigroup \(S_m = \langle m, m+1, m+2, \ldots, 2m-1 \rangle\).

The Graver basis of a numerical semigroup \(S\) is a collection of trades (congruences of factorizations in \(S\)) that allow for efficient solving of computationally difficult knapsack (optimization/resource management) problems, giving a (sometimes) polynomial-time algorithm of degree equal to the cardinality of the Graver basis. We present results that fully characterize these bases and their eventually quasipolynomial behavior in the case of three-generated semigroup, and discuss our attempts to adapt this methodology to semigroups with arbitrarily many generators.

We will begin by constructing a hexagonal lattice \(\Lambda_H\) and examining some of its properties. We then will examine shells of polytopes that have unfoldings that embed into \(\Lambda_H\). Such possible shells will be explored, and the simple case of tetrahedral structures will be discussed.

In this talk we will continue our exploration of groups that are generated by the reflections in the sides of a triangle with angles \(\pi/k, pi/m, \pi/\) for integers \(k,m,n\). In particular, we will introduce some aspects of 2-dimensional hyperbolic geometry to better understand these triangle groups in the case when \(1/k+1/m+1/k<1\). Then we will classify triangle groups by the geometry on which they act - spherical, Euclidean, or hyperbolic. Lastly, we will take a look at what can be said about generalizations in higher dimensions.

The Kunz cone is a geometric object defined by a set of inequalities of the form \(x_i + x_j \geq x_{i+j}\) where subscripts are in \(\mathbb Z_m\). Setting some subset of those inequalities equal, we obtain a face of the cone. In this talk, we will explore the face structures of the 4-, 5-, and 6-dimensional cones and we will also characterize faces of codimension 2 for any Kunz cone.

Minimal presentations of numerical semigroups give insights into the structure of factorizations of elements of that numerical semigroup. We discuss characterizing the achievable minimal presentation cardinalities of numerical semigroups by multiplicity and embedding dimension. We produce these characterizations through the relationship between a numerical semigroup's poset and its minimal presentation.

Kieran Hilmer, James Howard, and Anthony Park will each give a 20 minute talk as practice for the upcoming USTARS conference.

A numerical semigroup is a subset of the natural numbers that is closed under addition, contains zero, and has finite complement. Numerical semigroups with a fixed smallest positive element each correspond to an integer point in a geometric object called the Kunz cone. By studying the geometric properties of the Kunz cone, we are able to count the number of numerical semigroups that share certain properties. Using partial-fraction decomposition and constant-term evaluations, we are able to find the cross-sectional volume of the Kunz cone, which determines the growth rate of the number of numerical semigroups with fixed smallest positive element by number of gaps.

A numerical semigroup is a subset of the non-negative integers that contains zero, is closed under addition, and has finite complement in the non-negative integers. The smallest positive element of a numerical semigroup \(S\) is called the multiplicity of \(S,\) and the number of gaps of \(S\) (that is, the number of integers that are not in \(S\)) is called the genus of \(S\). Numerical semigroups of fixed multiplicity \(m\) are in bijection with certain integer points in a rational polyhedron \(P_m\) called the Kunz polyhedron. It is known that the number of numerical semigroups with genus \(g\) and fixed multiplicity \(m\) is eventually quasipolynomial, and it was conjectured that the starting place of quasipolynomial behavior is given by \(\binom{m-1}{2}\). In this talk will discuss where this comes from geometrically and what specific questions are still open.

Maryam: Today we will be introducing affine semigroups, specifically affine semigroups of dimension 2. We will look at factorizations of semigroup elements, trades, and discuss minimal presentations. We will also introduce shifted affine semigroups with fixed offsets to try and find patterns of their minimal presentation cardinality.

Gabbi: We will be discussing the quasipolynomial associated with the Frobenius number of semigroups with linear terms. This includes a brief demonstration of how the coefficients for the quasipolynomial can be solved for. This will serve as an introduction into research of the behavior of the Frobenius number when the elements of a semigroup are non-linear.

Hanh: In algebraic combinatorics, a numerical semigroup is a set of non-negative integers that is closed under addition, contains zero, and has finite complement. For a numerical semigroup \(S\) with multiplicity \(m\), the Kunz Poset of \(S\) is a partially ordered set obtained by mapping the elements of the Apery poset of \(S\) onto their equivalence classes in \(\mathbb Z_m\). This construction is of interest because it allows one to study the combinatorial properties of \(S\) using algebraic tools. In this presentation, we will explore the connection between the Kunz Poset and hyperplanes in \(S\), specifically how the Kunz Poset can be used to compute the face dimension of \(S\). This relationship has important implications for understanding the geometric structure of numerical semigroups and their applications in coding theory and algebraic geometry.

Caleb: An affine semigroup is a finitely generated additive submonoid of a Euclidean space, and it has many important applications in various fields of mathematics, such as combinatorics and algebraic geometry. The factorization problem asks whether every affine semigroup can be expressed as a product of irreducible affine semigroups, and if so, how many such factorizations exist. I will discuss the properties of affine semigroups, including their generators and relations, and how they can be factored into irreducible components, as well as the behaviors of their factorization as we increase the dimension of vector spaces.

"There are only two things certain in life: death and taxes" as the saying goes. Taxation affects all aspects of our life beyond just filing our annual income tax returns. It also affects and is affected by politics and macro and micro economic policy. Our goal in this talk is to explore some of these aspects of taxation based on audience feedback. Please come with questions!