Time:  Tuesdays 6:00pm  7:00pm 
GMCS 405 
Advisor:  Christopher O'Neill 
Organizers:  Cole Brower 
James Howard 
Fall Semester 2021 
Sep 14, 2021  Mariah Moschetti  From factorization graphs to zonotopes: a look at some geometry in numerical semigroup theory 
Abstract
In this talk, we revisit the poset of disjoint supports on \(k\) elements  a tool for counting the edges in factorization graphs of numerical semigroup elements. First, a brief overview of salient topics in numerical semigroup theory will be given. Then, we will explore the poset of disjoint supports on \(k\) elements and its geometric interpretations. We end with a connection to an interesting family of polytopes called zonotopes.


Sep 21, 2021  Cole Brower  Triangulations and CUDA 
Abstract
Triangulations give us a way to express potentially complicated convex polytopes as the union of simplices. Using this representation, one can extract important quantities such as the lattice points and volume by leveraging well known properties of simplices. Here, I will discuss an algorithm for computing triangulations as well as explain how we plan to accelerate this algorithm by leveraging GPUs and CUDA programming.


Sep 28, 2021  Jackson Autry  Exploring numerical semigroups through rays in Kunz polyhedra 
Abstract
The Kunz polyhedra are a family of polyhedra which have a close connection to numerical semigroups. These geometric objects offer a new viewpoint for studying numerical semigroups and grouping them into families. We study groups of numerical semigroups corresponding to rays in a Kunz polyhedron, and show how this view has led to new results.


Oct 5, 2021  Joe McDonough  Geometric approaches to studying numerical semigroups 
Abstract
The Kunz polyhedra and group cone are two related families of rational polyhedra that are intimately connected to numerical semigroups. In this talk we will introduce these polyhedra and how they are used to study properties of individual numerical semigroups, as well as some counting problems relating to numerical semigroups.


Oct 12, 2021  James Howard  The size of Graver bases of numerical semigroups 
Abstract
The Graver Bases of a numerical semigroup is a set of trades that allows us to move between factorizations while minimizing the number of trades required to do so. They are, however, generally very large and can be quite costly in terms of computing time. We explore shifted families of semigroups and conjecture a linear relationship between the size of their Graver bases.


Oct 19, 2021  Judy Wu  Factoring polynomials in numerical semigroup algebras 
Abstract
In this talk, we will discuss the reducibility of polynomials in numerical semigroup algebras. To begin, we will review the concepts of numerical semigroups, Frobenius numbers, and polynomial reducibility. Then, we will extend our knowledge to the reducibility of polynomials over finite fields. Ultimately, we introduce the central theme of our research: comparing factorization in semigroup algebras and the polynomial rings that contain them.


Oct 26, 2021  Andrew Morris  What To expect when your numerical semigroup is expecting: expected behavior of random numerical semigroups 
Abstract
We examine the expected value of the embedding dimension of random numerical semigroups with a fixed multiplicity. We approach this problem using geometric properties of the Kunz polyhedron and the numerical semigroup correspondence.


Nov 2, 2021  Kaelia Okamura  A twist on shifted families of numerical semigroups 
Abstract
In this talk, we will review shifted families of numerical semigroups. Next, inspired by what has already been done with shifted families of numerical semigroups, we will introduce a new family of semigroups and look into its poset structures and Frobenius number.


Nov 9, 2021  Cole Brower  A group cone safari: finding extremal rays in the group cone 
Joe McDonough 
Abstract
A numerical semigroup is a subset of the natural numbers that is closed under addition, contains zero, and has finite complement. The group cone provides an injection between numerical semigroups of a fixed smallest generator and integer points in a convex polyhedron. This approach has recently seen a lot of success providing a geometric interpretation of numerical semigroup invariants. We investigate the unexpectedly complicated structure of the extremal rays of these polyhedra.


Nov 16, 2021  James Howard  Examining and developing an algorithm for the arrangement and transcription of music notation into guitar tablature 
Abstract
In this talk, we explore the development of a discrete optimization algorithm that takes midi/music notation as input, arranges it (in the compositional sense, preserving the necessary harmonic and melodic structure), and outputs guitar tablature subject to playability constraints. We will cover the algebraic setup necessary to define our maps, as well as the music theory necessary to inform our arrangement algorithm, culminating in a vision of such an algorithm that could be implemented in a relatively computationally efficient way.


Nov 23, 2021  No Meeting, Thanksgiving Week  
Nov 30, 2021  TBA  TBA 
Spring Semester 2021 
Jan 26, 2021  Gilad Moskowitz  The structure theorem for sets of lengths for numerical semigroups 
Abstract
Let \(S\) be a numerical semigroup, then we define two additional numerical semigroups \(S_M\) and \(S_m\) which are constructed from the generators of \(S\). We describe relationships between these numerical semigroups and the original semigroup \(S\), and how we can use these to prove the Structure Theorem for Sets of Length for Numerical Semigroups, and it's components.


Feb 2, 2021  Nils Olsson  Music and fractals 
Abstract
Fracticality is a widely applied measure in areas of math (e.g. in dynamical systems) but can have applications in areas where we least expect, including in the study of music. We briefly explore music theory and some of the ways in which we can measure fracticality in music.
(With time permiting) Stationeers is an earlyaccess spaceengineering simulation game that provides an interesting microcosm in which math and physics can be applied for potential educational purposes. We briefly discuss a recent applied thermodynamics project of mine.


Feb 9, 2021  Panelists  Ph.D Program Panel 
Abstract
We will be joined by Jackson Autry, Misha Kutzman, Eduardo Torres Davila, and Isabel White, who are close friends of the SGPs seminar, to talk about their graduate school experience(s).


Feb 16, 2021  Mariah Moschetti  Manifolds  a topological introduction 
Abstract
Manifolds are ubiquitous in mathematics, comprising many of the surfaces we are familiar with, such as spheres. However, the formal definition of a manifold can be a difficult to digest for those who are not wellacquainted with pointset topology. In this talk, we parse through the formal definition of an \(n\)manifold, providing examples to illustrate each of the key topological ideas.


Feb 23, 2021  Cole Brower  Triangulations and CUDA (4:30pm start) 
 4:30pm  
Abstract
Triangulations give us a way to express potentially complicated convex polytopes as the union of simplices. Using this representation, one can extract important quantities such as the lattice points and volume by leveraging well known properties of simplices. Here, I will discuss an algorithm for computing triangulations as well as explain how we plan to accelerate this algorithm by leveraging GPUs and CUDA programming.


Mar 2, 2021  Tara Gomes  Hangin' out with Betti 
Abstract
In this talk we introduce a family of graphs that can be used to identify the Betti numbers and minimal trades between generators of a numerical semigroup. We then expand our notion of Betti numbers to include higher order relations, and reexamine these graphs viewed as simplicial complexes. Finally, we introduce the framework of free resolutions as a tool for capturing the full set of Betti numbers and homology of a numerical semigroup.


Mar 9, 2021  Brittney Marsters  A gentle introduction to connections between algebra and geometry 
Abstract
In this talk, we aim to introduce the audience to the ideas behind the correspondence of algebra and geometry, a perspective that gives us twice the tools and twice the questions. We will introduce the audience to some fundamental objects in algebraic geometry, including vanishing set of polynomials, affine varieties, radical ideals, and vanishing ideals. Lastly, for an algebraically closed field \(K\), we will discuss the bijective correspondence between radical ideals in \(K[x_1,...,x_n]\) and affine varieties in \(A^n\) to hint at the existence of meaningful connections between algebra and geometry.


Mar 16, 2021  Cole Brower and  The Kunz polyhedron and delta sets of numerical semigroups 
 4:30pm   Joseph McDonough 
Abstract
A numerical semigroup is a subset of the natural numbers that is closed under addition, contains zero, and has finite complement. The Kunz polyhedron provides bijections between numerical semigroups of a fixed smallest generator and integer points in a convex polyhedron. This approach has recently seen a lot of success providing a geometric interpretation of numerical semigroup invariants. We seek to utilize the Kunz polyhedron to better understand the "maximum of the delta set" of numerical semigroups.

Mar 23, 2021  Gilad Moskowitz  Sylver coinage and the Frobenius coin problem 
Abstract
In this talk I will give a very brief review of the Frobenius Coin Problem, and describe how it is related to the game Sylver Coinage. Then I will discuss Sylver Coinage, and the struggle for finding the best strategy in a given position. Finally, I will discuss the approach that I am using to try and find an optimized strategy. For anybody that is interested in playing the game before I present, you can visit this site (which I will talk about during the presentation).


Mar 30, 2021  No Meeting, Rest and Recovery Day  
Apr 6, 2021  Kiley Sprigg  A geometric approach to block monoids 
Abstract
The block monoid \(\mathcal B(G)\), over an Abelian group \(G\), is the set of all zero sum sequences of \(G\). Factorization length is one of several factorization invariants that provides an algebraic framework through which the structure of block monoids is commonly studied. In this thesis, we introduce a way to study the structure of block monoids from a geometric perspective. We look at max factorization length and explore current results for \(\mathcal B(\mathbb Z_4)\), \(\mathcal B(\mathbb Z_5)\), and \(\mathcal B(\mathbb Z_6)\), and present several conjectures for block monoids over larger Abelian groups.


Apr 13, 2021  Brittney Marsters  Optimizing polytopal norms with respect to numerical semigroups 
Abstract
Fix a polytope \(P\). The polytopal norm of a point with respect to \(P\) is the smallest dilation factor \(t\) such that \(tP\) contains this point. A numerical semigroup \(S\) is a subset of the nonnegative integers that contains zero and is closed under addition. Elements of \(S\) can be expressed as linear combinations of the generators of \(S\) where coefficients are taken to be nonnegative integers. To each of these expressions, we associate a point that we call a factorization of this element in \(S\). During this talk, we will discuss optimizing polytopal norms defined on sets of factorizations of elements of numerical semigroups. We will present results classifying the eventually quasilinear relationship for max and min polytopal norms for rational polytopes of dimension \(k\).


Apr 20, 2021  Mariah Moschetti  Edges, loops, and hypercubes: an exploration of the topological structure of graphs in numerical semigroup theory 
Abstract
Numerical semigroups are of great interest in factorization theory due to their highly nonunique factorizations. Given an element \(n\) in a numerical semigroup, various graphs can be constructed using the multiple factorizations of \(n\) as vertices. In this talk, we explore the topological structure of the factorization support graph of \(n\). We first characterize the number of edges in this graph and then use this characterization to show that the rank of its fundamental group is a quasipolynomial in \(n\).


Apr 27, 2021  Nils Olsson and  Atomic density of \(M_{4,6}\) and \(M_{6,10}\) 
Derek Rawling 
Abstract
We cover our recent exploration into quantifying the atomic densities of two particular arithmetic congruence monoids: \(M_{4,6}\) and \(M_{6,10}\); the process involved and what makes it challenging.

Fall Semester 2020 
Sep 2, 2020  Tara Gomes and  Minimizing minimal presentations 
Eduardo Torres Davila 
Abstract 
Paper
The Kunz polyhedra, \(P_m; m \in \mathbb N\), are a family of rational polyhedra whose integer points are in bijection with numerical semigroups of multiplicity \(m\) with faces in correspondence with a family of finite posets (called Kunz posets). In this talk we introduce a new mathematical property  the minimal presentation of a Kunz poset. We then explore this property to arrive at two main results. First, we find a combinatorial solution to the problem of computing face dimension. We then prove minimal presentation size and structure to be an invariant within the faces of \(P_m\) and arrive at a new combinatorial algorithm for computing the minimal presentation of a numerical semigroup.


Sep 9, 2020  Gilad Moskowitz  Machine learning for use in predicting the Mobius function in large integers 
Abstract
In this talk, we introduce the Mobius function and some properties of machine learning. We describe the idea of a problem being NPHard (but not NPcomplete) and explain how predicting the Mobius function with enough accuracy using machine learning can demonstrate that the "Factorization Problem" is solvable in polynomial time.


Sep 16, 2020  Cole Brower and  Length density of numerical semigroups 
Joseph McDonough 
Abstract
A numerical semigroup is a subset of the whole numbers, closed under addition, with many wonderful properties which have been the subject of extensive, ongoing, investigation. We propose a new property, "length density", and provide many interesting results.


Sep 23, 2020  Meeting Cancelled  
Sep 30, 2020  Mariah Moschetti  Exploring the edge density of graphs from factorization sets of numerical semigroups 
Abstract
An important property of numerical semigroups is that their elements may have multiple factorizations. Given an element \(n\) in a numerical semigroup \(S\), graphs can be constructed using the multiple factorizations of \(n\) as vertices. In this talk, we explore two such graphs: the minimal presentation graph of \(n\) and the factorization graph of \(n\). While there are many interesting questions one can pose, this talk will focus on questions of edge density. That is, "How far are these graphs from being complete?" As it turns out, this question has surprising connections to topology and are describable using quasipolynomials.


Oct 7, 2020  Kiley Sprigg  A geometric approach to block monoids 
Abstract
The block monoid \(B(G)\), over an abelian group \(G\), is the set of all zero sum sequences of \(G\). Factorization length is one of several factorization invariants that provides an algebraic framework through which the structure of block monoids is commonly studied. In this talk we introduce a unique way to study the structure of block monoids from a geometric perspective. We explore current results for \(B(Z_4)\) and present current conjectures/open problems for larger Abelian groups.


Oct 14, 2020  Brittney Marsters  Polytopal norms and numerical semigroups 
Abstract
A polytopal norm for a given polytope \(P\) and a vector \(\vec{a}\) is defined to be \(\ell_{P}(\vec{a})= \min\{t: \vec{a}\in tP\},\) where \(tP\) denotes the \(t^{\text{th}}\) dilation of \(P\). In this talk, we will discuss optimizing polytopal norms defined on \(\mathsf Z_S(n)\) for a fixed numerical semigroup \(S\). We will present recent results classifying the eventually quasilinear relationship for max polytopal norms in two dimensions, and discuss current musings pertaining to three dimensions.


Oct 21, 2020  Nils Olsson and  Atomic density of regular arithmetic congruence monoids is zero 
Derek Rawling 
Abstract
Arithmetic congruence monoids (ACMs) are multiplicative monoids \(M_{a,b}\) consisting of all positive integers \(n\) satisfying \(n \equiv a \bmod b\). A nonunit \(x \in M_{a,b}\) is irreducible/an atom if it cannot be factored as a product of nonunits within \(M_{a,b}\), and the atomic density of \(M_{a,b}\) is the limiting ratio of atoms to all elements of the ACM. In this paper we show that regular ACMs \(M_{1,b}\) have atomic density equal to zero.


Oct 28, 2020  Gilad Moskowitz  The structure theorem for sets of lengths for numerical semigroups 
Abstract
Let \(S\) be a numerical semigroup, then we define two additional numerical semigroups \(S_M\) and \(S_m\) which are constructed from the generators of \(S\). We describe relationships between these numerical semigroups and the original semigroup \(S\), and how we can use these to prove the Structure Theorem for Sets of Length for Numerical Semigroups, and it's components.


Nov 4, 2020  No Meeting, Election Week  
Nov 11, 2020  No Meeting, Veterans Day  
Nov 18, 2020  Mariah Moschetti  Counting with posets  numerical semigroups to cubical complexes 
Abstract
In this talk, we revisit factorization graphs of numerical semigroups. Let \(n\) be an element of a numerical semigroup \(S\). The factorization graph of \(n\) in \(S\) is the graph which has the set of factorizations of \(n\) as its vertex set and edges between any two vertices when the factorizations share at least one copy of the same generator of \(S\). While trying to count the edges in these graphs, we construct a poset that captures the structure of our counting argument. The Möbius function will be introduced as a tool to analyze this poset. In our analysis, a surprising geometric interpretation will arise.


Nov 25, 2020  No Meeting, Thanksgiving  
Dec 2, 2020  Tara Gomes  A survey of the Kunz polyhedra 
Abstract 
Paper 1 
Paper 2 
Paper 3
Recently much attention has been given to a family of rational polyhedra (\(P_m, m \in \mathbb N\)) whose integer points lie in bijection with numerical semigroups with smallest generator \(m\). In this talk we give a thorough introduction to the Kunz polyhedra, explore a combinatorial interpretation of their faces, and ask ourselves just how much geometry has to say about the structure of a numerical semigroup.


Dec 9, 2020  Cole Brower and  The Kunz polyhedron and delta sets of numerical semigroups 
Joseph McDonough 
Abstract
A numerical semigroup is a subset of the natural numbers that is closed under addition, contains zero, and has finite complement. The Kunz polyhedron provides bijections between numerical semigroups of a fixed smallest generator and integer points in a convex polyhedron. This approach has recently seen a lot of success providing a geometric interpretation of numerical semigroup invariants. We seek to utilize the Kunz polyhedron to better understand the "maximum of the delta set" of numerical semigroups.

Spring Semester 2020 
Feb 5, 2020  Tara Gomes  An introduction to the Kunz Polyhedron 
SDSU  
Feb 12, 2020  Zachary Dickinson  A Groebner talk 
SDSU 
Abstract
In the 1960s, the idea of Groebner bases were invented and since then have gone on to being applied to solving nonlinear system of equation to assisting in statistic analysis. This talk is designed to introduce the idea of Groebner bases from its origins of polynomial division of more than one variable to the definition of the Groebner basis itself. We discuss a way of computing the Groebner basis of an ideal with respect to some monomial ordering. We then mention that a Groebner basis can be unique, up to monomial orderings, of which there are finitely many. Lastly, we offer the listener a geometric representation of the collection of all Groebner bases of an ideal in the form of a fan.


Feb 19, 2020  Erin Grooms  Maximal factorization lengths of affine semigroups 
SDSU 
Abstract
For numerical semigroups, the maximal factorization length function has a linearity property that allows us to find the maximal factorization length of an element, \(M(w)\), by using the maximal factorization length of the element minus the multiplicity, \(M(wg_1)\). In this presentation, I will discuss this phenomenon and extend it to affine semigroups of dimension two with three generators.


Feb 26, 2020  Brittney Marsters  An overview of factorization invariants 
SDSU  
Mar 4, 2020  Isabel White  Shifted affine semigroups 
SDSU  
Mar 11, 2020  Eduardo Torres Davila  Minimal presentations of Kunz posets 
SDSU 
Abstract 
Paper
Several recent papers have examined a rational polyhedron \(P_m\) whose integer points are in bijection with the numerical semigroups (cofinite, additively closed subsets of the nonnegative integers) containing \(m\). A combinatorial description of the faces of \(P_m\) was recently introduced, one that can be obtained from the divisibility posets of the numerical semigroups a given face contains. We extend the notion of a minimal presentation of a numerical semigroup to posets, and use it to determine the dimension of the corresponding face.


Mar 18, 2020  No Meeting, COVID19 Insanity  
Mar 25, 2020  Gilad Moskowitz  Properties of maximum and minimum factorization length 
SDSU  in numerical semigroups  
Abstract 
Slides
The maximum and minimum factorization lengths are known to be quasilinear for sufficiently large \(n\). In this presentation I will discuss a generalized definition of the Apery set that we used to find a formula for the quasilinear portion of the factorization lengths.


Apr 1, 2020  No Meeting, Spring Break  
Apr 8, 2020  Mariah Moschetti  The fundamental group of factorization graphs 
SDSU 
Abstract
The fundamental group of a topological space is one of the simplest invariants to study in algebraic topology. The fundamental group is a quotient group constructed from the equivalence classes of loops in topological space. Two loops are considered equivalent if one can be continuously deformed into the other. In this talk, we explore the fundamental groups of familiar spaces, such as the sphere, circle and torus. Additionally, we will investigate the fundamental group of a connected graph and determine that its rank depends on the number of edges in a spanning tree. This brings us to an important connection to semigroup theory: for any numerical semigroup \(S\), we can construct a graph for each element whose vertices are factorizations and whose edges are determined by a certain choice of trades between them. The research question for this project is whether the rank of the fundamental group of these graphs is eventually quasipolynomial. As of now, the edges that are in or not in these graphs (i.e., the graph's edge density) has been the main area of focus to answer this question.


Apr 15, 2020  Kiley Sprigg  Block monoids 
SDSU  
Apr 22, 2020  Tara Gomes  Wilf's Conjecture as a Combinatorial Game 
Apr 29, 2020  Zach Dickinson  Practice Thesis Defense 
Abstract
A numerical semigroup \(S\) is an additive subsemigroup of the positive integers. The family of shifted numerical semigroups \(S(n)\) is found by adding a positive integer \(n\) to each generator of \(S\). Each numerical semigroup is associated to a homogeneous toric ideal, constructed from the minimal relations of its generators, while every toric ideal associates with a finite set of reduced Groebner basis. The Groebner fan of an ideal is the geometric representation of the set of all reduced Groebner bases of the ideal. In this thesis, we algorithmically characterize Groebner fans of toric ideals of 3generator shifted numerical semigroups when \(n\) is sufficiently large. We then use this algorithm to prove the number of facets of the Groebner fan and the number of generators in the universal Groebner basis are quasilinear as functions of \(n\).


May 6, 2020  Isabel White  Practice Thesis Defense 