Jonathan Mousley

About Me

My name is Jonathan Mousley. I am a second-year Mathematics PhD student at Duke University. I am currently supported by an NSF Graduate Research Fellowship. I graduated with a Bachelor of Science in Mathematics and minor in Mechanical Engineering from Utah State University in spring 2022. As an undergraduate, I participated in research relating to graph theory, computational linear algebra, and geometry processing. I have interest in Topology, Applied Mathematics, and their intersections.

Email:
Office: Gross Hall 304
Calendly: jonathanmousley (schedule a meeting with Jonathan)

Research

As an Undergraduate Research Fellow at Utah State University, I studied spectral methods for optimal graph layouts and labelings of directed graphs under Dr. David Brown and Dr. LeRoy Beasley. Most of my time was spent studying (2,3)-cordial labelings of directed graphs, a labeling scheme that intuitively implies balance in directed graphs. For more information, see publications.

At the University of Michigan-Dearborn REU Site in Mathematical Analysis and Applications in the summer of 2021, I studied a model for phase retrieval with applications to optical microscopy, a computational imaging technique that involves reconstructing a high-resolution image from many low-resolution images. My mentors were Dr. Aditya Viswanathan and Dr. Yulia Hristova.

During summer 2021, I participated in research at the Summer Geometry Institute at the Massachusetts Institute of Technology. I participated in projects on discrete analogues to operators on smooth surfaces, 3D image reconstrution from 2D images, and anisotropic Schrödinger Bridge Optimal transport. My project mentors were Dr. Amir Vaxman (Utrecht University), Dr. Noah Snavley (Cornell University), and Dr. Justin Solomon (MIT).

Publications

Cordiality of Digraphs.

Beasley, L., Mousley J., Santana M., Brown D.
Journal of Algebra Combinatorics Discrete Structures and Applications 10 (2023), 1-13.

abstract arXiv

A (0,1)-labelling of a set is said to be friendly if approximately one half the elements of the set are labelled 0 and one half labelled 1. Let g be a labelling of the edge set of a graph that is induced by a labelling f of the vertex set. If both g and f are friendly then g is said to be a cordial labelling of the graph. We extend this concept to directed graphs and investigate the cordiality of sets of directed graphs. We investigate a specific type of cordiality on digraphs, a restriction of quasigroup-cordiality called (2,3)-cordiality. A directed graph is (2,3)-cordial if there is a friendly labelling f of the vertex set which induces a (1,−1,0)-labelling of the arc set g such that about one third of the arcs are labelled 1, about one third labelled -1 and about one third labelled 0. In particular we determine which tournaments are (2,3)-cordial, which orientations of the n-wheel are (2,3)-cordial, and which orientations of the n−fan are (2,3)-cordial.

(2,3)-Cordial Oriented Hypercubes.

Mousley, J., Beasley L., Santana M., Brown D.
Submitted as conference proceedings at SEICCGTC 2021.

abstract arXiv

In this article we investigate the existence of (2,3)-cordial labelings of oriented hypercubes. In this investigation, we determine that there exists a (2,3)-cordial oriented hypercube for any dimension divisible by 3. Next, we provide examples of (2,3)-cordial oriented hypercubes of dimension not divisible by 3 and state a conjecture on existence for dimension 3k + 1. We close by presenting the only 3D oriented hypercubes up to isomorphism that are not (2,3)-cordial.

(2,3)-Cordial Trees and Paths.

Santana, M., Beasley L., Mousley J., Brown D.
Submitted as conference proceedings at SEICCGTC 2021.

abstract arXiv

Recently L. B. Beasley introduced (2,3)-cordial labelings of directed graphs in [1]. He made two conjectures which we resolve in this article. He conjectured that every orientation of a path of length at least five is (2,3) cordial, and that every tree of max degree n=3 has a cordial orientation. We show these two conjectures to be false. We also discuss the (2,3) cordiality of orientations of the Petersen graph, and establish an upper bound for the number of edges a graph can have and still be (2,3) cordial. An application of (2,3) cordial labelings is also presented.

Visualizations

Schrödinger Bridge Optimal Transport (Left) vs. Anisotropic Schrödinger Bridge Optimal Transport (Right)

Interpolation on the left is dictated by the geometry of the surface alone. On the right, anisotropy tensors (visualized by ellipses) make local modifications to the metric of the surface and influence the transport interpolation. Intuitively, along long axes of ellipses, the surface is "pinched" and along short axes of ellipses, the surface is "stretched." The algorithms used to produce these visuals were designed collaboratively by me, Juan Atehortúa, Faria Huq, Adrish Dey, and Dr. Justin Solomon at SGI 2021. Special thanks to Dr. Keenan Crane who created this mesh.

Teaching

I’m passionate about teaching mathematics. During undergrad, I worked as both a tutor and recitation instructor for my department. I worked as a recitation instructor for Math 2250: Linear Algebra and Differential Equations for six consecutive semesters and Math 2210: Multivariable Calculus for one semester. I have been assigned as a teaching assistant for Math 111L: Labratory Calculus at Duke University for both Fall 2022 and Fall 2023.

Course	Title	Term	Role	Institution
Math 111L	Laboratory Calculus I	F23	Teaching Assistant	Duke University
Math 111L	Laboratory Calculus I	F22	Teaching Assistant	Duke University
Math 2250	Linear Algebra and Differential Equations	Spr22, F21, Spr 21, F20, Spr 20, F19	Recitation Instructor	Utah State University
Math 2210	Multivariable Calculus	F21	Recitation Instructor	Utah State University

Outreach

I served as the Academic Senator for the College of Science at Utah State University. I ran on an agenda to expand the visibility of undergraduate research opportunities on campus and to secure funding for our student-run peer mentorship program. To accomplish these goals, I recruited nearly 30 undergraduates to serve on our student council. While in office, I worked with this council to hold several research themed events, including two Rapid Fire Research presentation nights and an REU Application Preparation Workshop.

Personal

My hobbies include reading (math and non-math related), hockey, all things coffee related, and (more recently) photography. I also have a one-year-old mini-labradoodle named Euler who can't ever get enough attention.