Research & Teaching

Friday series 1 Friday teaching

Saturday series 2 Saturday teaching

Saturday series 3 Saturday series 4

#### Research talks

**Friday 3:00-5:00 pm : Series I, Westminster, Gore 133**

**3:00** - Greg Vogel

**Title:** Discrete Event Branching Process Model of Memory CD4+ Memory T cell differentiation.

**Abstract:** Much work has been done looking at CD4+ Memory T cells, our research isolates the individual events that occur to CD4+ T cells throughout an immune response. Memory CD4+ T cells are the cells that orchestrate a response to a pathogen upon re-challenge, and the memory T cell repertoire is created during primary and subsequent infections or through vaccination. Understanding this generation can lead to better developments of vaccines and a greater understanding of immune responses to infections. We generate a Discrete Event Branching Process to represent the events that occur during the development of the CD4+ T cell lineage. We are looking to understand the probability distributions associated with each step that occurs over the course of development, and then determine the variation from traditional deterministic models. We then use Python to run simulations to track a CD4+ T cell throughout an immune response over a 400 hour interval, receiving constant antigenic stimulation. We track over each time step the number of naive CD4+ T cells, effector CD4+ T cells, Memory CD4+ T cells, and total CD4+ T cell count. Our simulation yields results to show the variation from deterministic curves of an immune response along with overlaying the curves determined by the mean of all trials and plus or minus one standard deviation, and we also calculate the mean number of mitoses a cell will have before undergoing apoptosis, which supports the observation of different responses to the same pathogen in different individuals, and can be used to understand the parameters that exist when considering treatments to pathogens such as vaccine development.

**3:20** - Jianlong Han, Southern Utah University

**Title:** A numerical approximation for a nonlocal evolution equation with Kac potentials

**Abstract:** We propose a stable, convergent finite difference scheme to solve numerically a nonlocal evolution equation which describes the continuum limit of Ising spin systems with Glauber dynamics and Kac potentials. We also prove that the scheme is uniquely solvable and the numerical solution will approach the true solution in the L^infty norm.

**3:40** - William Cocke, Brigham Young University

**Title:** Nonequivalent subsets of lattices in 2 and 3 dimensions

**Abstract:** The use of Pólyas Theorem in crystallography and its application has greatly simplified many counting and coloring problems. Previous work has been done which establishes a formula for the number of translationally nonequivalent periodic subsets of a lattice. The number of nonequivalent subsets of the lattice can be solved using Pólya counting on the group of relevant symmetries acting on the lattice. When equivalency is defined via a superlattice, the use of Pólyas Theorem is equivalent to knowing the cycle decomposition of the action of the group elements on the quotient lattice. Algebraic methods are used to determine the cycle decomposition of all possible isometries of the lattice in two or three dimensions, including rotation, reflection, and all combinations thereof. Explicit formulas are demonstrated for all two and three dimensional symmetries. A list of possible isometries group in two dimensions is provided based on the superlattice relationship to its parent lattice.

**4:00** - Derek Hein, Southern Utah University

**Title:** Decompositions of lambda K_n using multigraphs on four vertices with edge frequencies only 1, 2 and 3

**Abstract:** We show how to decompose lambda copies of the complete graph (for the minimal value of lambda) into so--called LOE--type, OLE--type, LEO--type and ELO--type multigraphs.

**4:20** - Skyler Simmons, Brigham Young University

**Title:** Numerical investigations of the rhomboidal 4-body problem

**Abstract:** I describe various numerical methods used to investigate the existence and stability of the rhomboidal 4-body problem. In particular, I outline a least-squares method of finding initial conditions of the system of ODEs governing the behavior of the orbit, and present a method which can be used to determine the stability of the resulting solution.

**4:40** - Geoff Bradway, University of Utah

**Title:** Homology of data sets

**Abstract:** A topic of interest in classical topology is guring out the essential structures of a space, such as a surface or metric space, using homology theory. To examine the structure of data sets, a collection of discrete points, instead of decomposing an object into simplexes, you use the discrete points to create simplexes, at a scale alpha. In this research project, I examined both the 0-dimensional and 1-dimensional homologies of data sets, over a range of alphas, using a computational approach. The primary focus was using computational methods to fully examine and understand low dimensional homologies in order to better understand the data at hand. The programs created allow the user to take a set of data and get the precise H0 and H1 homologies at any alpha (both the total number of basis elements as well as representative for each basis) , plot the number of basis for the H0 and H1 homologies verses alpha, and many other minor features (such as examining the non-trivial H0 homologies, or plotting the graph of the data set at hand).

**Saturday 8:30-11:00 a.m. : Series II, SLCC, N185**

**8:30** - David Tate, Utah State University

**Title:** Full isolation number: some extremal results

**Abstract:** A set of nonzero entries of a (0,1)-matrix is an *isolated set* if no two entries belong to the same row, no two entries belong to the same column, and no two entries belong to a submatrix of the form [1 1; 1 1]. The isolation number of a matrix is the maximum size over all isolated sets. The isolation number of a matrix is a well-known and well-used lower bound for the matrix's Boolean rank. We will discuss the isolation number of the adjacency matrix of various graphs and develop some extremal results for *n *x* n* matrices with isolation number *n*.

**8:50** - Scott Roy, Utah State University

**Title:** On the Boolean rank of tournament matrices and in particular rotational tournament matrices

**Abstract:** An *n*-tournament is an orientation of the complete graph on *n* vertices. A *tournament matrix* is the adjacency matrix of an* n*-tournament, equivalently, an *n* x *n* (0,1)-matrix *M* satisfying *M*+M^T = *J - I*, where *J* is a matrix of all 1s. If there is an arc from* v* to* u* in the tournament we say *v beats u*. An *n*-tournament is *doubly regular* if each pair of vertices together beat* k* common vertices; necessarily, each vertex beats 2*k*+1 vertices and *n* = 4*k*+3. A *rotational n-tournament *is a doubly regular tournament with vertices {0,1,2,..., *n-1*} and* i* beats *j* if* j - i *≡ *s *modulo *n*, where* s* is an element from a carefully chosen *symbol set S* with |*S*| = (n-1)/2, 0 not in *S*, and, for any *x,y* in *S*, |*x -y*| ≠0 modulo *n*. We discuss the directed biclique cover number of tournaments, equivalently the Boolean rank of their adjacency matrices, and highlight some tools which provide insight into these parameters. In particular, we develop results about the Boolean rank of rotational tournament matrices, and structural implications having full Boolean rank has on tournament matrices.

**9:10** - Sarah Mousley, Utah State University

**Title:** Maximum number of high-scoring players in round robin tournaments

**Abstract:** Consider a round robin tournament among *n* contestants (every contestant competes with every other) and no ties are accepted. To a Graph Theorist this is an orientation of a complete graph on *n* vertices or an *n-tournament*. Suppose all contestants who beat less than *m* other contestants are eliminated. What is the maximum number of contestants that can remain after the elimination? Equivalently, what is the maximum number of contestants that can beat at least *m* others? Call this number *f*(*m,n*). We completely describe *f*(*m,n*) and several other related functions, as well as how these functions give the size of the largest induced *m*-regular tournament which can be present in an *n*-tournament.

**9:30** - Sebrina Cropper, Utah State University

**Title:** The structure of the *n*-tournament poset

**Abstract:** An *n*-tournament is an orientation of the complete graph on *n* vertices. The *score* *s*_{i} of a vertex *v*_{i} of *n*-tournament *T* is the number of vertices *x* with an arc from vertex *v*_{i} to *x* in *T*. The score vector of an *n*-tournament is the nondecreasing list of scores (*s*_{1},s_{2},..., s_{n}) which satisfies sum_i s_i_{ }= n choose 2. If *T*_{n} is the set of all *n*-tournaments, a partial order (a reflexive, symmetric and transitive ordering) of the elements of *T*_{n} can be obtained using the score vectors. We investigate this partially ordered set and, among other things, identify some structure of the poset that corresponds to the *n*-tournaments which are strong (vertices are mutually reachable) and those which are not.

**9:50** - Jonathan Franklin, Utah State University

**Title:** Isolation number and lower bounds for Boolean rank and intersection number

**Abstract:** Suppose *M* is a (0,1)-matrix. The *isolation number* of *M* is the largest number of entries equal to 1 in *M* such that no two ones are in the same row, no two ones are in the same column, and no two ones are in a submatrix of *M* of the form [1 1;1 1]. The isolation number is a well-known (possibly best-known but not best) lower bound for the Boolean rank of *M*, or, equivalently, if *G* is the graph corresponding to *M*, the biclique cover number of *G*. The Boolean rank is known to be an NP-Complete problem, and the isolation number is similar in difficulty. We develop some theory giving structural relationships which yield optimal results for the isolation number of certain classes of directed graphs, in particular, restricted classes of tournament matrices.

**10:10** - Christopher Thatcher, Utah State University

**Title:** Chicken McNuggets, computers, generating functions, oh my!

**Abstract:** At some point in history McDonalds sold Chicken McNuggets in quantities of 6, 9, and 20. Suppose we wanted to give the cashier an impossible task and order the maximum number of McNuggets that cannot be obtained as a combination of 6, 9, and 20. It turns out 43 is this number. This is a special case of a well-known problem in Number Theory called the Frobenius number problem. Formally, given a set {*a*_{1}, a_{2}, ..., a_{k}} of positive integers, what is the maximum number which cannot be expressed as* c*_{1}a_{1}+c_{2}a_{2}+^{...}+ c_{k}a_{k}, where each *c*_{i} is a nonnegative integer? We will show how to solve the Frobenius number problem via generating functions, survey the (sad) current state of the art of the Frobenius number problem, and showcase an algorithm which solves the problem for any given set of integers.

**10:30** - David E. Brown, Utah State University

**Title:** Intersection number of graphs and matrix ranks

**Abstract:** Suppose you have a finite set *S* from which you take subsets *S*_{1}, S_{2}, ..., S_{n} and make a graph representing the intersection relationships of the sets. That is, you form the *intersection graph G *whose vertices are the sets {*S*_{i}} and vertices *S*_{i} and *S*_{j} are adjacent in *G* if and only if *S*_{i }∩ S_{j} ≠ emptyset. From the opposite perspective, suppose you have a graph *G*; now determine a collection of sets *S*_{1}, S_{2}, ..., S_{n} from a superset *S* so that *G* is the intersection graph of {*S*_{i}}. Now minimize the cardinality of *S*; this minimum cardinality is the* intersection number of G*, an idea introduced by Erd?s, Goodman, and Po?a circa 1966. I will showcase equivalent definitions of the intersection number via matrix factorizations over anti-negative semirings, such as the non-negative integers and the Boolean algebra (where the only numbers are 0 and 1 and all operations are as in the reals except 1+1=1). I will also pose some new problems on intersection representations which involve a tolerance; e.g., *G* is the *k-tolerance intersection graph* of *S* if *G*'s vertices are represented by subsets of *S* and vertices are adjacent if and only if their sets' intersection has size at least *k*.

**Saturday 8:30-11:00 a.m. : Series III, SLCC, N195**

**8:30** - Sum Chow, Brigham Young University

**Title:** An inverse source problem for elastic wave propagation in metamaterials

**Astract:** In recent years, many metamaterials have been constructed in laboratories. These materials have many interesting and unexpected properties, with many potentially useful and important applications. In this talk we will discuss an inverse source problem connected with the study of wave propagation in some of these materials. More specifically, we will discuss results showing that it is not possible to use a finite number of frequencies to solve the source identification problem but using a continuous interval of frequencies will uniquely identify the source. Some error bounds for finding approximate source will also be developed.

**8:50** - Mahmud Akelbek, Weber State University

**Title:** On the subdominant eigenvalues of stochastic matrix

**Abstract:** For a primitive stochastic matrix S, upper bounds on the second largest modulus of an eigenvalue of S are very important, because they determine the asymptotic rate of convergence of the sequence of powers of the corresponding matrix. In this talk, I will provide an attainable upper bound on the second largest modulus of eigenvalues of a primitive matrix that make use of the scrambling index. I will also present some of the new results on the scrambling index and generalized scrambling index of primitive digraphs.

**9:10** - Emma Turner, Brigham Young University

**Title:** The k-S-rings and k-characters of finite groups

**Abstract:** The *k*-characters of a finite group *G* were defined by Frobenius in 1896. The 1-characters of *G* are the well known irreducible characters of* G* and correspond to representations of the group algebra of *G*. The 1-*S*-ring is the center of the group algebra and determines the 1-characters. The *k-S*-rings, for *k *>1, are a generalization of the 1-*S*-ring, motivated by the definitions of the *k*-characters of *G*. We will discuss some of my research on 2-,3-, and 4-*S*-rings, including results on commutative 3-*S*-rings, the character theory of the 2-*S*-ring of *G* and its relationship to the 2-characters of *G*, and when the *k-S*-ring determines the group.

**9:30** - Seth Armstrong, Southern Utah University

**Title:** A nonstandard discretization method for numerical analysis of a Lotka-Volterra food web model

**Abstract: ** We compare the standard and non-standard discretizations to show why the latter is preferred to the former. A uniquely solvable, stable, semi-implicit finite-difference scheme is proposed for this system that converges to the true solution uniformly in a finite interval. A simple Hopf bifurcation arises for the system. This is based on work done with J. Han.

**9:50**- William Cocke, Brigham Young University

**Title:** Web Representations of the Symmetric Group

**Abstract:** The irreducible representations of the symmetric group were constructed by Alfred Young using tableaux structures from combinatorics. An alternative construction was presented by Kazhdan and Lusztig using Kazhdan-Lusztig polynomials and left cells. These representations have a canonical basis which has applications in Lie theory. There is another construction for irreducible representations of *S*_{3n} from combinatorial objects called webs. Calculations in this setting are significantly simpler than calculations in the corresponding Kazhdan Lusztig representation. We compare the combinatorial properties of these representations to those of the Kazhdan-Lusztig representations. Equivalency of the canonical bases is established for the corresponding representations of *S*_{3}, S_{6}, S_{9}, S_{12}, and* S*_{15}. A counterexample is presented for *S*_{18}_{ }demonstrating the failure of equivalency for larger symmetric groups.

**10:10** -Amanda Francis, Brigham Young University

**Title: **The concavity axiom in FJRW quantum singularity theory

**Abstract: **Mirror symmetry is a phenomenon from physics that has inspired a lot of interesting mathematics. In the Landau-Ginzburg setting, we have two constructions that we can associate to an affine singularity with a group of symmetries. These are the FJRW ring and the orbifold Milnor ring. Both are vector spaces equipped with multiplication and a pairing (making them Frobenius algebras). The multiplicative structure of the FJRW Ring is determined by a generating function which requires the calculation of certain numbers, which we call correlators. One of the methods we have for calculating these correlators is the so-called concavity axiom. In my talk I will discuss the FJRW construction, correlators, and the methods we are developing to implement the concavity axiom.

**10:30** - Chelsea Kennedy, Brigham Young University

**Title:** Quadratic total character groups

**Abstract:** The *total character τ *of a finite group *G* is defined to be the sum of the irreducible characters of *G*. In 1997, K. W. Johnson posed the following question concerning total characters: For a finite group *G*, do there necessarily exist an irreducible character χ and a monic polynomial *f*(*x*) in [*x*] such that *f*(χ)=* τ, *where *τ* is the total character of *G*? If such a character and polynomial exist and the polynomial is degree 2, we say that *G *is a *quadratic total character group*. In this talk we will discuss examples of quadratic total character groups, and show that if a *p*-group is a quadratic total character group, it is extraspecial.

**Saturday 8:30-11:00 a.m. : Series IV, SLCC, N224**

**8:30** - Taylor Sibbett, Westminster College

**Title:** Discrete Signal Processing with Fractal Wavelet Transforms

**Abstract:** Digital signal processing is used to represent functions (signals, data) efficiently using some optimal combination of basic functions. By taking advantage of the structure of information inherent in typical signal data, complicated functions that represent images, sounds, and other signals can be represented with less information. However, it is difficult to understand what sort of underlying structure exists with a given signal, so different discrete wavelet transforms are used on different data sets to obtain more efficient representations. In my research, I have implemented numerous fractal wavelet transforms, which uses the self-similarity of certain fractals to match the self-similarity used in wavelets. My research advisor and I are continuing to experiment with new data sets and comparing the efficiencies of different algorithms. In this talk, I will discuss selected fractal wavelet transforms that I have implemented, and present various numerical results.

**8:50** - Andrew Larsen and Kali Wickens, Westminster College

**Title:** Beyond Regression: Using Learning Machines to Predict NBA Performance

**Abstract:** Each year, NBA teams make million-dollar investments by drafting collegiate and international prospects in the NBA draft; it is therefore critical to be able to predict the future performance of those players. While current prediction models rely on regression and other basic statistical techniques to make these predictions, the use of artificial intelligence has not been explored in this context. Learning machines, a branch of artificial intelligence, have shown marked improvement over their counterparts in making predictions in other fields, such as handwriting recognition, spam filters, and even movie ratings. To make predictions on the future success of NBA players, we utilize two types of learning machines: support vector machines and decision trees. We train both the support vector machines and learning trees on collected data of past collegiate basketball players to create models which can then classify new players coming into the model. These machines make better predictions of player performance than previously established baselines.

**9:10** - Dane Bartlett, Southern Utah University

**Title**: Predator-Prey Equation with Limited Resources

**Abstract: **We study the stability of equilibrium solutions to a system involving two species with limited resources. We also develop a stable numerical scheme to verify theoretical results of the system.

**9:30** - Michael Bentley, University of Utah

**Title:** Cloaking for the Maxwel l Equations using Active Sources

**Abstract:** We present a method for cloaking ob jects from probing electromagnetic fields using active sources. The sources are a few point-like multipolar devices, so the ob ject is not completely isolated from the exterior. Numerical simulations verify that these active sources do not produce large electromagnetic fields far away from the cloaked region, and cancel most of the probing field within the cloaked region.

**9:50** - Nozomu Okuda, Brigham Young University

**Title:** Metamaterials and Waves: Problems in Mathematical Modeling and Computational Simulation

**Abstract:** The application of metamaterials can be as fantastic as invisibility cloaks or as mundane (and nevertheless useful) as better cell phone service. These can be accomplished because of the unusual way metamaterials interact with electromagnetic waves. To better understand these relationships, simulations based on mathematical modeling can be used. Although the models of wave propagation through normal materials are well established, the models of wave propagation through metamaterials are not. Thus, the challenge of ensuring correctness remains, since strange discrepancies between simulation and prediction either indicate that there are shortcomings and false assumptions embedded within the underlying mathematical model or else intimate that there are characteristics and phenomena not included in the overarching scientific doctrine.

**10:10** - Zach Boyd, Brigham Young University

**Title:** Harmonic mappings with SIFD and new minimal surfaces

**Abstract:** Planar harmonic functions $f=h+overline{g}$ are generalizations of analytic functions. Univalent planar harmonic functions can be constructed by shearing a specific class of univalent analytic function with certain analytic dilatation. Most results in this area have discussed families of functions whose dilatations are finite Blaschke products. In this paper, we present new collections of harmonic mappings with dilatations that are singular inner functions and discuss some their properties. In some cases, these mappings lift from the complex plane onto minimal surfaces in 3-space and appear to be new minimal surfaces with peculiar properties.

**10:30** - Brandon Wiggins,

**Title:** Preliminary Studies of the Relationship Between the Two-Point Correlation Function and Random Walk Step-length

**Abstract:** Primordial black holes (PBHs) form from large fluctuations in the radiation density field of the early universe. The spatial distribution and clustering of PBHs are of special interest to the cosmological community as PBHs could serve to seed supermassive black holes and intimately probe cosmology and general relativity. Obtaining a realization of such a distribution directly from the correlation function for PBHs, however, becomes computationally impractical due to the extreme rarity of the perturbations which would produce a collapse to singularity in the early universe. In this paper, we present preliminary results of an alternate, random walk scheme in 3 space and aim to reproduce the spatial distributions of PBHs. Using simulation and applying two-point correlation function estimators, we investigate the nature of the mapping between the set of all probability density functions for step length and the set of resulting correlation functions. We provide discussion on algorithm details and theory and compare our results with the literature.

#### Teaching talks

**Friday 3:00-5:00 p.m., Westminster, Gore 133**

**3:00** - Troy Goodsell, BYU-Idaho

**Title:** Extraction of Roots in Sir Isaac Newtons’ The Method of Fluxions and Infinite Series.

**Abstract:** In this talk we will look at Newton’s use of examples as a teaching tool in his Method of Fluxions. We will specifically look at his use of extraction of roots dealing with polynomial examples.

**3:20 **- David Brown, Utah State University

**Title:** Common Statistical Formulas: Don't Memorize 'em; Derive 'em

**Abstract:** I present a (certainly known, but evidently not well-known) linear algebraic derivation of the formulas for mean, covariance, and correlation. I argue these derivations yield a geometric understanding of the formulas affording better understanding. These derivations use the ideas of projections onto subspaces, and the geometry behind the linear algebra illuminates the meaning of the formulas.

**3:40** - Said Bahi , Southern Utah University

**Title:** Dimensions of Service Quality in Higher Education: Underlying Structure of Students’ and Faculty Perception

**Authors:** Saïd Bahi, Azmi Ahmad and Deanna Dillard

**Abstract: ** The purpose of this work is to examine the perception of undergraduate students at a regional public university of quality service in higher education. In studying the perceived dimensions of quality teaching and programs, we investigated the differences in perception of students and faculty. A survey was conducted in which each respondent was asked to rate various indicators of quality teaching and quality programs on a scale of 1 to 5. For logistics and convenience reasons, the population studied was limited to business and sciences undergraduate students and their respective faculty. This paper reports and tests dimensions measuring service quality in higher education and discusses what dimensions students value in their educational experience.

**4:00** - Damon Bahr, BYU

**Title:** It Takes a Village: Investigating the Critical Role Clinical Faculty Play in Mathematics Teacher Education

**Abstract**: This presentation will report the results of a study that examined the influence of clinical faculty in the developing dispositions of preservice teachers towards reform-based mathematics teaching and learning. Specifically, we studied the relationship between the dispositions of two groups: (a) preservice elementary teachers involved in field-based practica, and (b) three types of clinical faculty, who support these preservice students’ practica experiences. Results provide evidence for the existence of meaningful relationships between changes in the dispositions of preservice teachers in field practica and their perceptions of the dispositions of the clinical faculty with whom they work.

**4:20** - David Stowell, Brigham Young University - Idaho

**Title:** Polynomial and Nonlinear Eigenvalue Problems in Photonics

**Abstract**: Nonlinear eigenvalue problems arise in the modeling of many phenomena and is an area of active research. In this talk we discuss two examples: one is a polynomial eigenvalue problem while the other is a purely nonlinear eigenvalue problem. Solution methods for each are discussed.

**Saturday 8:30 – 11:00 a.m., SLCC, N222**

9:10 - Amanda Cangelosi, University of Utah

Title: Using the Common Core State Standards Integrated Secondary Curriculum to Motivate a Teacher Enrichment Course

Abstract: As the Common Core State Standards begin to be implemented in schools, teacher education courses are presented with a new opportunity for enrichment. We take a first look at a two-semester course for current Utah secondary teachers that uses the Integrated Secondary Mathematics I and II Common Core State Standards as a guide for discussion of advanced mathematical topics, as well as pedagogical methods. We summarize our course structure and content, highlight activities, and share outcomes and reflections.

9:30 - Emina Alibegovic, University of Utah

Title: Transformational geometry in grades 6-12

Abstract: The Utah Core pays considerably more attention to transformations in its geometry portion than it has before. We will discuss the content of the core and the opportunities this approach affords for development of connections with other topics. We will also discuss some problems students might encounter when learning about transformations.

9:50 - Violeta Vasilevska, Utah Valley University

Title: Origami/GeoGebra Projects for Geometry Classes

Abstract:

In this talk we discuss two increasingly popular teaching tools in Geometry:

- Origami (the art of paper folding), and

- GeoGebra (dynamic geometry software).

In particular, we present few origami-math and GeoGebra projects that can be used in geometry classes to enhance students’ learning. These projects are easily understood by undergraduate students and offer a great opportunity for hands-on, discovery-based learning.

10:10 - Brynja Kohler, Utah State University

Title: Implementing Lesson Study at the Park City Mathematics Institute

Abstract: Over the past two summers I worked with teams of teachers in the Secondary School Teachers Program at the Park City Mathematics Institute on implementing the model of Japanese Lesson Study. In this talk, I will summarize our process and share an overview of the two lessons that resulted from our work: One lesson focuses on teaching logical reasoning, and in particular, addresses conditional statements; the second lesson was designed to develop students’ understanding of solutions to systems of equations. Both lessons were tested with high school sophomores.

10:30 - Paul R. Hurst, BYU-Hawaii

Title: Long Distance Teaching

Abstract: We will discuss the various pieces of technology involved in successful long distance teaching. This enables students at two separate locations to be taught simultaneously. One application is that mathematics departments at different universities can coordinate with each other to offer a more diverse variety of courses for their students while simultaneously cutting costs. Another application is that it allows students that need to travel for an internship or family event to “attend” class while they are away.