## Come to our Weekly Colloquium

Colloquia are generally held on selected Tuesdays throughout the term, starting at 4pm. Talks generally last about 50 minutes, with 10 minutes for questions at the end. Students, faculty, staff, and the mathematical public are all cordially invited to attend. Speakers, titles, and abstracts are updated throughout the semester as details become available.

You can contact the colloquium committee at colloquium-math@jmu.edu.

##### Fall 2024

4pm, Roop 103

Michael Robert, Virginia Tech

**Abstract: **TBA

##### Spring 2024

4pm, Roop 103

Showcasing work by student researchers in the Department of Mathematics and Statistics. Details below.

Speaker: **Michael Glover **

Title: Navigating Geodesics: Exploring Shape Transformation Through Dynamic Simulations in MATLAB

Abstract: This presentation explores the dynamic conservation of geometric properties—area in 2D and volume in 3D—as triangles, quadrilaterals, and tetrahedra undergo transformations along certain geodesic paths. Utilizing MATLAB for computational modeling to determine the positions of shapes through time, and OpenSCAD to visually represent these transformations by lofting sequential 2D shapes into a cohesive 3D model, this research highlights the conditions under which geometric conservation is maintained. By merging mathematical analysis with practical simulation tools, this study offers insights into the stability and transformative behaviors of shapes, yielding implications for theoretical insights and practical applications alike.

Speaker: **Natalie Huey**

Title: Reinforcement Rates and Their Impact on Mathematical Models of Warfare

Abstract: In this talk, we will examine two types mathematical modeling that predict the outcome of a military conflict. After focusing in on Lanchester’s Combat models, we vary initial conditions, reinforcement rates, and types of military forces to create new scenarios with interesting results. Our main finding was between guerrilla forces, where there is a certain number of reinforcements that will ensure a reinforced army will always win, no matter the length of reinforcement period.

**Monday**, 4pm, Roop 103

What the Research Says about Active Learning – and What it Doesn’t

Estrella Johnson, Department of Mathematics, Virginia Tech

**Abstract: **The research literature in undergraduate mathematics education, and other science and engineering education fields, is showing a general convergence towards the position that actively engaging students during classroom instruction improves learning and persistence outcomes. In this talk I will provide an overview of the preponderance of research on the use of “active learning” in undergraduate STEM courses, calling specific attention to the consensus findings of this research. I will also discuss what we know about how instructor attitudes, local supports, and departmental environments are related to increased usage of active learning. However, in order to contextualize these findings, I will also draw out the nuances and considerations that challenge the oversimplified idea that “any active learning is better for all students”.

4pm, Roop 103

Inclusivity in the Classroom

Beth Cochran, Department of Mathematics and Statistics, JMU

**Abstract: **Inclusivity in the classroom -- ensuring students feel welcome, seen, heard, and valued. Ideas and strategies for small changes that can be easily implemented to promote an inclusive classroom environment for students. Topics will include facilitating connections, inclusive syllabus, inclusive language, and inclusive & lower-cost course materials. This talk was presented at the National Organization for Student Success (NOSS) Culture of Care Summit, February 2024.

**Friday, 4:10pm, **Roop 103

Assessing Dengue Risk Globally Using Non-Markovian Models

**Abstract: **Dengue is a vector-borne disease transmitted by Aedes mosquitoes. The worldwide spread of these mosquitoes and the increasing disease burden have emphasized the need for a spatiotemporal risk map capable of assessing dengue outbreak conditions and quantifying the outbreak risk. Given that the life cycle of Aedes mosquitoes is strongly influenced by habitat temperature, numerous studies have utilized temperature-dependent development rates of these mosquitoes to construct virus transmission and outbreak risk models. In this study, we advance existing research by developing a mechanistic model for the mosquito life cycle that accurately accounts for the non-Markovian nature of the process.

**Wednesday, 4:30pm, **Roop 103

What is the Galois group of a random, large degree number field?

**Abstract: **Work establishing the quadratic equation, cubic equation, and quartic equation dates back to antiquity and concluded with Cardano’s Ars Magna around 1545. Verification that there is no quintic equation, i.e. no general formula expressible in radicals for the zeros of a degree five polynomial, had to wait for Abel and Ruffini in the 1820s. A decade later, Galois introduced his eponymous theory to determine which high degree (> 4) polynomials can have their zeros expressed in radicals, despite there being no general procedure to do so. In modern terms, Galois theory associates a permutation group (the Galois group) to the roots of a polynomial which can be used determine solvability. Remarkably, a consequence of a theorem of Hilbert is that 100% of the time, the Galois group of a “random" degree n, monic, irreducible polynomial with integer coefficients is S_n, the full group of permutations on n letters. That is, generically, one does not expect restrictive relations between the roots of a polynomial.

Since one can attach a number field (a finite field extension of the field of rational numbers) to a monic, irreducible, polynomial over the integers (namely its splitting field), one might then ask if a "random" degree n field extension is likely to have Galois group S_n. While the answer can be extracted for small n from various known results, this talk will address this question for large n. We prove, assuming a conjecture of Bhargava and some mild assumptions, that the density of degree n, S_n-extensions among all degree n extensions is 0 as n goes to infinity. That is, one should not expect the Galois group of a large degree extension to be S_n. Similarly, we also prove that extensions with Galois group S_t-wreath-G with t << log(n)^2 has density 1 among all degree n extensions as n goes to infinity. In this way, we characterize the groups most likely to appear as the Galois group of a random number field of high degree.

I will mention some previous work with Matt Friedrichsen and ongoing work-in-progress with Jiuya Wang.

**Monday, **4pm, Roop 103

How quickly does heat decay in infinite bodies?

**Abstract: **Understanding how heat diffuses in an environment is a natural physical question. Moreover, heat diffusion can be modeled by a differential equation whose study leads to very interesting mathematics, even in more abstract settings. In this talk, we will look at how heat decays in infinite spaces, and, in particular, how heat decay is influenced by boundary condition. In many cases, understanding heat decay requires studying certain harmonic functions. We will look at both heat decay and these harmonic functions in many examples, and the talk concludes with some open questions.

4pm, Roop 103

Integrating Computing into Mathematics Education: A Case of Python Programming in Combinatorial Tasks

Elise Lockwood - Dept. of Mathematics, Oregon State University/NSF

**Abstract: **Computational activity, and programming in particular, comprise an increasingly essential aspect of scientific activity, and engaging in computing is as accessible as it ever has been. In mathematics education, there is a need to investigate the ways in which students’ computational activity can support their reasoning about mathematical concepts. In this talk, I present results from a study in which undergraduate students engaged with Python programming tasks designed to elicit particular combinatorial ideas. I highlight noteworthy aspects of students’ experiences with computing in this mathematical context, including benefits and drawbacks of working in a computational environment. I suggest that even for students with little programming experience, the computational environment supported their combinatorial reasoning in valuable ways. I frame this study within my prior work, and I address practical issues related to implementation and discuss pedagogical implications. Overall, I seek to frame these specific findings about Python programming in mathematics as an instance of a broader phenomenon, namely highlighting the ways in which computing may be leveraged to support students’ engagement with scientific concepts and practices. I conclude by framing this work within my broader research program.

##### Fall 2023

4pm, Roop 103

Connecting Microbial Dots

Laura Tipton, Biology Department (joint appointment in M&S), James Madison University

**Abstract: **Microbiomes, the collection of all the microbes in an environment and their abiotic conditions, have been a hot topic in human and environmental health for the last decade. To go beyond a descriptive list of bacteria present requires pulling methods from biology, ecology, statistics, machine learning, graph theory, and more. Spanning multiple microbiomes, this talk explores different ways to examine connections both within microbial communities and between microbes and their environments.

##### Spring 2023

4pm Roop 103

Student Research Talks

Student Research Presentations

May 3, 4pm, Roop 103

Speakers: **Matthew Caulfield** and **Aidan Chadha**

Title: Predicting the Jet Boundary of a Turbulent Coanda Wall Jet Using Experimental Test Results

Abstract: The Coanda effect is the tendency of a fluid to stay attached and follow the curvature of a rounded surface. Turbulent Coanda wall jets, often seen in aeronautical and aerospace applications, utilize this effect to enhance flow lift and deflection, and change the location of jet breakaway points. Jet breakaway is the scenario in which the fluid no longer follows the curvature but instead travels tangential to the surface. The relationship between a Coanda jet’s fundamental characteristics (such as slot width and operating pressure), its acoustics emission, and the location of jet breakaway is not widely understood. In particular, it is argued that if better predictions of Coanda jet noise were available, the benefits associated with the Coanda effect would be more widely realized. For example, it has been observed that curved jets are often noisier than similar non-curved jets. The work presented herein attempts to rectify the lack of knowledge in this area; specifically addressing the issue of how the Coanda jet boundaries are influenced by jet operating characteristics. A model describing the jet boundary as a function of slot width (w) and operating pressure (p) is determined based on recent experimental data. Predications are then compared with additional experimental results and conclusions drawn. The method of Least-squares Optimization is also applied to the data to improve this model, perhaps yielding more accurate agreement with measurements.

Speaker: **Evan Lawing**

Title: Determining the Factors that have an Influence on Movie Gross

Abstract: In the ever-changing movie industry, predicting the next blockbuster movie is the main goal of every production and distribution studio. There are many factors that can cause a movie to be a flop or a success. . Due to the risk in the investment being made and the ever-changing preferences of the viewer, production and distribution studios work tirelessly to try to find out which factors will make their movie the next blockbuster. In this project, we will be analyzing the effects of different variables on film domestic gross. These variables include MPAA rating, genre, days in theater, and budget of the film among others. To keep up with current viewer preference trends a dataset consisting of 567 films from the years 2016-2019 and 2021 was constructed. Linear regression analysis was used to help answer multiple questions about domestic gross in relation to different factors.

Speaker: **Andre Mas**

Title: Fourier Finite Element Methods for an Elliptic State Constrained Optimal Control Problem on Axisymmetric Domains

Abstract: Given an elliptic state constrained optimal control problem defined over an axisymmetric domain, Fourier series decomposition can be used to turn this 3D problem into a sequence of weighted 2D problems. We describe a Fourier finite element method for this problem, and display various numerical examples.

Speaker: **Max Misterka**

Title: Unsolved problems related to quantum walks on graphs

Abstract: In this talk, we will discuss some unsolved problems related to graph theory, linear algebra, and quantum computing. Recall that a graph is a collection of points (called vertices) and lines between the points (called edges). Loosely, a graph X admits *perfect state transfer* between two vertices u and v if a “quantum signal” that starts at u completely transfers itself to v after some positive time; X admits *uniform mixing* if a “quantum signal” that starts at an arbitrary vertex becomes evenly distributed among the vertices after a fixed positive time. (All of this can be expressed formally using linear algebra.) These two properties of graphs turn out to be pretty rare, and there are still many open questions about what kinds of graphs can admit perfect state transfer or uniform mixing. For example, Gabriel Coutinho conjectured in 2014 that no trees with more than three vertices admit perfect state transfer. A more surprising example: no one knows whether the cycle graph on 9 vertices (a relatively small graph) admits uniform mixing!

Speaker: **Colby Sherwood**

Title: A representation theoretical approach to the critical groups of hypercube graphs

Abstract: Hypercube graphs are an interesting class of graph which exhibit many regularity properties that make them useful in a variety of applications. An important algebraic invariant of hypercube graphs is the critical group. However, finding a formula for the critical group of hypercube graphs has remained an open problem for over 20 years. We introduce a new approach to this problem using the representation theory of symmetric groups. From this approach we derive a new method of calculating the critical groups of hypercube graphs that may be useful in finding a general formula.

4pm, Roop 103

Making Statistics Classes Interesting Using Software

Jonathan Fernandes, Department of Mathematics, University of Maryland, College Park

**Abstract: **We will be introducing some ongoing work on pedagogy of introductory statistics courses at the University of Maryland. We will be to show how some 'difficult' topics can be covered without intimidating students (with the math), and perhaps in many cases inspiring them to feel excited about the subject at hand.

**Note: **This is a Monday colloquium

4pm, Roop 103

The Math That Simplifies Data and AI

Hala Nelson, Department of Mathematics and Statistics, James Madison University

**Abstract: **I will survey the math that necessarily underlies many seemingly disparate real world AI and data projects. Many people trying to enter the data and/or AI fields get overwhelmed by the amount of information that is out there. Math in this case serves a simplifying and unifying purpose, arming both newcomers and seasoned practitioners with a solid foundation that helps them build robust systems and assess existing ones efficiently. In this sense, math opens the black-box that many refer to in the data community. We will unify machine learning models and natural language models under one mathematical structure, showcase geospatial analysis and the integration of different kinds of data into useful formats, while at the same time stressing the importance of the quality of data and the downfalls of bad data. If time allows, I will briefly describe our own experience working with Harrisonburg City's emergency and infrastructure services, and/or discuss how to help our students transition seamlessly from STEM to the high-tech industry.

4pm, Roop 103

Transforming Institutional Practices through Multidisciplinary Data Science programs for Research and Education

Padmanabhan Seshaiyer, Department of Mathematical Sciences, George Mason University

**Abstract: **Institutions of higher education all across the globe have been focusing on integrating data science into a wide range of university curricula through programs building from foundations in mathematics, statistics and computer science with applications to real-world problems arising from sectors including business, industry, government, health, education and much more. In this talk, we will describe examples of sustainable data science programs and pathways for developing innovative multi-disciplinary research and education opportunities to solve societal challenges. We will also discuss potential opportunities to collaborate on new developments in data science to engage the next generation workforce in the Commonwealth.

##### Fall 2022

4pm, Roop 103

Joshua Ducey, Mathematics & Statistics, James Madison University

Some Open Research Problems In Graph Theory and Algebraic Combinatorics

**Abstract:** I will survey some of my favorite open problems in graph theory and algebraic combinatorics, including: the “missing Moore graph” problem; a problem concerning an invariant of the n-dimensional hypercubes; and others involving subsets and subspaces. The mathematical background required to understand and work on most of these is minimal, and a lot of experimentation can be done with a computer.

Some past and current students of mine have made progress on several of these. Students interested in getting involved in mathematics research are especially encouraged to attend.

4pm, Roop 103

Project-Based Learning for MATH 103

Eva Strawbridge, Mathematics & Statistics, James Madison University

**Abstract: **MATH 103 falls under the Cluster 3 General Education (The Natural World) requirements for JMU. Specifically, these courses are meant to expose students to "scientific investigations into the natural world use analytical methods to evaluate evidence, build and test models based on that evidence, and develop theories." For Cluster 3 courses, the learning objectives are not dependent on subject specific mastery of content but rather on cognitive skills such as problem solving and methods of inquiry. Ultimately students should gain sufficient experience so that they are better prepared to consume and evaluate the validity of scientific information presented to them. In this presentation I will be discussing an approach for developing these critical reasoning skill sets through project-based learning as an applied math course.

4pm, Roop 103

A Tour of 3D Printed Mathematics: Tritangentless trefoils, ideal graph configurations, and chaotic attractors

Stephen Lucas & Laura Taalman, Mathematics & Statistics, James Madison University

**Abstract****:** In this talk we’ll take a 3D-printed tour of a collection of mathematical objects we have made over the years. We’ll start with building tritangentless trefoils and show how to make them roll as easily as possible. We will then look at ways to construct collections of ideal graph configurations in three dimensions. Finally, we will show how to create 3D mesh models to accurately visualize a variety of chaotic orbits. Along the way we will explore the technical and computational tools needed for creating 3D-printable mathematical models, including OpenSCAD, Mathematica, and MATLAB, and provide resources to help students and mathematicians who wish to create their own models.

4pm, Roop 103

**Abstract**: Factoring large integers is a hard problem and all known algorithms run in (sub)-exponential time. The security of modern cryptosystems (like RSA) depends on factoring remaining computationally hard. In the eighties, Lenstra and Lenstra showed how one can use elliptic curves from number theory to factor large-ish integers. We explain the method and outline its limitations (don’t worry, RSA with ’n’ more than 200 digits is still quite secure). While the talk is intended to be as self-contained as possible---we will define elliptic curves, for instance---it would be helpful to know long division, modular arithmetic and how to compute the derivative of an implicit (polynomial) function.

Ravi Shankar, Mathematics & Statistics, James Madison University

##### Spring 2022

3:10 via zoom

Acoustic Modeling of the Rocket Flame Trench at Wallops Island Flight Facility

Abby Maltese, research student, Mathematics and Statistics, James Madison University**Abstract**: When a launch vehicle lifts off, its exhaust is guided away from the launch vehicle by a flame trench. It has been found that certain flame trench geometries can add to the acoustic stress on the rocket which can damage the launch vehicle and payload (Ranow, 2021). This study aims to add to the understanding of how flame trench geometry affects the sound emitted by the rocket exhaust as it flows through the flame trench at launch. These discoveries can help guide future flame trench design.

This study focuses on the MARS Pad 0A flame trench at the Wallops Island Flight Facility. Key assumptions are that all the flame trench materials have the same properties and will remain intact during launch. Additionally, it is assumed that the rocket exhaust flow is steady and does not change over time. These assumptions are made due to the scope of this project and the emphasis on analyzing the geometry of the flame trench and its relationship to acoustic pressure opposed to the materials or change of flow over time.

The flame trenches were modeled in Fusion360 and then analyzed and visualized in COMSOL. Five different models were modeled and tested. After the first set of simulations were completed, the boundary conditions were altered, and the simulations were run again resulting in a more realistic scenario. This study examines how frequency and amplitude affect the resulting acoustic pressure throughout the flame trench and how the change in geometry alters the flow of the soundwaves. The importance of three-dimensional acoustic simulations are discussed.

Visually Representing Propositional Logic

Andre Mas, research student, Mathematics and Statistics, James Madison University**Abstract:** Propositional & predicate logic provides a standard for the formal reasoning we use in Mathematics- an argument is valid if we can formally prove that the conditions we impose imply the conclusion. Proofs are traditionally done symbolically, but is this the *only* way? We examine the

idea of expressive completeness in logic, which raises a visual representation of propositional logic and proofs in this system.

Incidence Matrices of Subsets and the Representation Theory of the Symmetric Group

Colby Sherwood, research student, Mathematics and Statistics, James Madison University**Abstract:** Incidence matrices describing the intersection of subsets of a set have been studied by mathematicians since the 1960s. They arise naturally in many combinatorial investigations, and the ranks of these matrices over finite fields have applications to design theory, coding theory and algebraic combinatorics. While the ranks of the inclusion matrices have been calculated, very little is known about the other obvious incidence relations such as having intersection of a fixed size. In this talk we show how the situation can be understood in terms of the representation theory of the symmetric group, and we solve the rank problem for 2-subsets vs. n-subsets intersecting in a set of size 1.

3:10 via zoom

Data To The Rescue: JMU Partnering With Harrisonburg City Fire Department

Max Aedo Espicto, Michael Michniak, Ashray Shah, Dominic Gammino, Chloe Powell, Colby Sherwood,

Reggie Wilcox, Mathematics and Statistics, James Madison University

**Abstract: **The Harrisonburg Fire Department is one of the city’s most valuable resources, and optimizing their performance is crucial in maintaining the safety, well-being, and livelihood of the citizens of Harrisonburg.

Recently, the Fire Department has obtained $4.9 million funding to build their fifth fire station. We, as students at James Madison University, have been working during Spring 2022 semester to help the fire department determine the optimal location for Fire Station 5 based on the department's incident and

response data. We joined the incident data provided to us with population, social vulnerability, and property value data. In this talk, we present our journey with the data, along with multiple approaches to locating the fifth fire station, including Monte Carlo simulations, machine learning techniques, and geospatial analysis. We also outline the next steps where data driven analysis will improve the performance of the Fire Department. This is an ongoing partnership, with both short and long term goals. The colloquium presentation at the Department of Mathematics and Statistics is our practice talk, before our presentation at the Fire Department Head Quarters on Thursday April 28.

This project is funded by the PIC Grant and by the Department of Mathematics and Statistics at JMU.

3:10 via zoom

Elliptic Curves and Differential Equations: Sending Codes or ODEs

James Sochacki, Mathematics and Statistics, James Madison University

**Abstract:** If you google ‘elliptic curve’, you will get some graphs and a general equation of an elliptic curve. You will also see that it is used in cryptography (secure communication techniques). In this talk, I will present a method of generalizing elliptic curves and secure communication techniques. This will involve modeling physical forces and conservation principles through Ordinary Differential Equations. The idea will be demonstrated through graphs and animations.

3:10 via zoom

Thicket Density

Siddharth Bhaskar, Computer Science, James Madison University

**Abstract: **A *set system* is a domain X along with an arbitrary family of subsets of X - we do not impose any particular structure on this family. Set systems are of interest in various disciplines such as discrete and computational geometry, machine learning theory, and more. Various combinatorial invariants can be associated with any set system; two examples are *VC dimension* and *Littlestone **dimension*. There is a precise sense in which the relationship between VC and Littlestone dimension corresponds to the relationship between learning via *non-adaptive* and *adaptive* queries.

One of the fundamental results in the combinatorics of set systems is the *SauerShelah lemma*, which says that a certain integer-valued function, the *shatter function*, grows either at most polynomially or at least exponentially depending on whether the VC dimension of the set system is finite or infinite. This has been called an "eigentheorem" for its many applications (to statistics, discrete geometry, graph theory, model theory) as well as many proofs (algebraic, combinatorial, etc.).

In this talk we shall introduce a new integer-valued function that we call the *thicket shatter function*, and show that it has exactly the same relationship to Littlestone dimension that the shatter function has to VC dimension. However, whereas the growth rate of the shatter function can assume any real value greater than 1, the growth rate of the thicket shatter function must be integer valued! This last fact does have a purely combinatorial proof (which I do not understand), but time permitting, I shall sketch a short proof that uses the notion of compactness from model theory.

3:10pm via Zoom

Recognize, Respond, Refer: JMU Faculty’s Role in Assisting Distressed Students

David Onestak, Counseling Center, James Madison University

**Abstract****: **This presentation provides an overview of collegiate mental health and the rationale for faculty engaging students around this topic. Information will be offered to help faculty better recognize the signs of student distress, respond in a helpful and effective manner, and refer students to campus resources that can support and assist them.

##### Fall 2021

3:10 via Zoom

A Novel, Flexible, Unified Framework for Survival Data

Prabhashi Withana Gamage, Mathematics & Statistics, James Madison University

**Abstract:** The proportional hazards (PH) model is, arguably, the most popular model for the analysis of lifetime data arising from epidemiological studies, among many others. In such applications, analysts may be faced with censored outcomes and/or studies that institute enrollment criteria. Censored outcomes arise when the event of interest is not observed but rather is known relevant to an observation time(s). The “enrollment issue” arises from studies that exclude participants who have experienced the event prior to being enrolled in the study. To analyze the aforementioned data, herein we propose a novel unified PH model that can be used to accommodate both of these features. To facilitate model fitting, an expectation-maximization (EM) algorithm is developed. To provide modeling flexibility, a monotone spline representation is used to approximate the cumulative baseline hazard function. The performance of our methodology is evaluated through a simulation study and is further illustrated through the analysis of two motivating data sets; one that involves child mortality in Nigeria and the other prostate cancer.

3:10 via Zoom

A Theory of Change to Practice: Using Qualitative and Quantitative Data to Drive Systemic Changes to Our Chemistry Curriculum

Benny Chan, The School of Science, The College of New Jersey

**The College of Science and Mathematics STEM+DEI Speaker**

**Abstract:** The changing demographics of the NJ student population towards more first generation and students of color was forecasted by decades of census data. Change is hard. Change is complex. Coupling the COVID-19 pandemic, we have seen a seismic shift in student preparation that requires a paradigm shift in our teaching. Luckily, The College of New Jersey’s School of Science was already engaged in doing systemic changes to our curriculum to make the courses more inclusive and student centered. We have developed a theory of change, the experimentalist teacher, to help to manage the current conditions and to anticipate the future changes to our student population. We have three pillars to the theory of change, gaining empathy and understanding of our students, a changed toolkit of acceptable pedagogy, and developing a common language, values, and understanding of our responsibility. We have gathered a tremendous amount of data about our most vulnerable student populations to help us design and assess a model for teaching general chemistry. The data informed work has driven additional work into our Inorganic, Analytical, Organic and Physical chemistry curricula and even to study issues like the sense of belonging and respect in our classrooms. The Chemistry Department has developed our vision and mission to be as inclusive as possible so that we can increase the numbers of successful and thriving students in our majors and chemical professionals.

3:10 via Zoom

Rain, Hail, and Drip Frames of the Schwarzschild-de Sitter Geometry

Tehani Finch, Physics & Astronomy, James Madison University

**Abstract:** The Schwarzschild spacetime geometry is the unique vacuum spherically symmetric solution of the Einstein equations of general relativity, and is used to model many sources of gravity, from planets to stars to black holes. Many coordinate systems for this geometry have appeared in the literature. Subsets of these coordinate systems are associated with observers moving inwardly along radial geodesics of this geometry. Such observers have been categorized as being in the “rain” frame, a “hail” frame, or a “drip” frame. This framework naturally progresses into a search for counterparts of these coordinate systems for other spacetime geometries. Notable examples include the geometry corresponding to a cosmological constant, de Sitter (dS) spacetime, and the spacetime that combines a spherical source with a cosmological-constant background, known as the Schwarzschild-de Sitter (SdS) spacetime. We find coordinate systems for the SdS geometry that turn out to differ from the naïve extrapolations of the Schwarzschild and dS geometries.

3:10 via Zoom

Pascal, Fibonacci, Polynomials and Differential Equations

Jim Sochacki, Mathematics and Statistics, James Madison University

**Abstract:** Pascal developed a triangle made up of natural numbers that contains the Fibonacci sequence in it. This is well known. However, what is not well known is that there are polynomials and their products in the Pascal triangle that solve some well-known differential equations. Through these differential equations one can develop more general Pascal type triangles that contain Fibonacci sequences in a straight forward manner. This talk will be accessible to anyone who has passed the first two semesters of a standard calculus sequence.

3:10 via Zoom

Donaldson's Diagonalizability Theorem

Guillem Cazassus, The Mathematical Institute, University of Oxford

**Abstract:** In 1983, Donaldson proved a striking theorem giving restrictions on the intersection form of a simply connected smooth 4-manifold. Together with results of Freedman, this implies the rather surprising existence of many topological 4-manifolds that do not admit smooth structures.

I will explain the ideas of its original and beautiful proof, which involves studying a space of solutions to a certain PDE (up to equivalence) that I will introduce: the self-dual instanton equation. These are generalizations of Maxwell's equations.

3:10 via Zoom

Varieties, Orbit Spaces, and the Heisenberg Group *H*_{3}

Andre Mas, research student, Mathematics and Statistics, James Madison University**Abstract:** A fundamental goal in mathematics is to understanding global actions on spaces. One way this may be done is by studying the maps induced on various related structures, such as the representation variety, which is constructed from the space’s fundamental group and a matrix Lie group. We will describe how this strategy is used to analyze the fixed point sets of three involutions that generate the mapping class group of a closed, connected, oriented, genus 6 surface. In the process, a case is made for the use of the 3×3 Heisenberg group in the study of representation varieties.

Efficient (*j*,*k*)-Domination on Chrysalises

William Nettles, research student, Mathematics and Statistics, James Madison University**Abstract:** Rubalcaba and Slater define an efficient (j, k)-dominating function on graph G as a function f : V (X) → {0, ..., j} so that for each v ∈ V (X), f(N[v]) = k, where N[v] is the closed neighborhood of v (Robert R. Rubalcaba and Peter J. Slater. Efficient (j, k)-domination. Discuss. Math. Graph Theory, 27(3):409-423, 2007). For regular graph G the set of efficient dominating functions is closely related to the (1)-eigenspace of G. A 3-regular caterpillar is a tree obtained from a path by adding a pendant vertex to every vertex of degree 2. A chrysalis is a 3-regular graph constructed by adding a cycle through the leaves of a 3-regular caterpillar. We characterize the planar chrysalises that admit efficient dominating functions, as well as the values j and k for which an efficient (j, k)-dominating function can be constructed. Towards extending our characterization to all chrysalises we characterize efficient domination on a class of non-planar chrysalises.