# JMU Math & Stat: Colloquium

## 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 2022

4pm, Roop 103

TBD

Eva Strawbridge, Mathematics & Statistics, James Madison University

4pm, Roop 103

A Tour of 3d Printed Mathematics

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

### Past Colloquia

##### 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.