Math Students Shine at Conferences

July 19, 2022
Gonzaga University News Service

SPOKANE, Wash. – Gonzaga University mathematics students won awards at two conferences for presenting research that with more development could lead to:

  • Creating efficient mail delivery systems for large offices and routes for delivery services.
  • Simulating tree breakage based on the curvature and material properties of the tree.
  • Predicting the orbits of satellites and asteroids with greater accuracy and efficiency.

Three of the students won top honors at the Pacific Inland Mathematics Undergraduate Conference at Washington State University in April: Luke Martin for his research talk – “Mobius Book Embeddings” – and Max Pansegrau and Sara Jane Lynn for their poster presentation – “Modeling Motion of Trees in the Wind.”

Ben Lombardi won a second-place poster award at the Pacific Northwest section of the Society of Industrial and Applied Mathematics conference at WSU’s campus in Vancouver in May. Most in attendance were graduate students.

“It was really inspiring to hear about how all sorts of abstract topics in math are being applied to real-world problems in industry,” Lombardi said. “Upon receiving second place for my poster I was pretty stunned.”

It Started in a Class

Martin, a senior studying computer science and math, got the research bug during a class.

“I had Dr. Tom McKenzie for Abstract Algebra I and really enjoyed the humor and passion he brought every day to lecture, so I decided to ask him if he had any research projects I could work on with him.”

“He and Dr. Shannon Overbay, along with two other students, Nick Linthacum and Lin Ai Tan, had been working on book embedding problems and he allowed me to join. I am extremely grateful that I was allowed to be a part of their group, as they had already put so much work into it when I joined.“

Indeed, the group collectively gave eight presentations on embedding at various conferences during the academic year.

Martin, of Rufus, Oregon, offered an overview of the group’s work that led to his research talk.

“Graphs are essentially just sets of vertices (dots) with lines (edges) connecting them,” he wrote. “The main problem we were dealing with was which graphs could be placed on different shapes without the lines crossing. Our group found that these graphs where the lines do not cross correspond to valid delivery system strategies, which has many real-world applications.”

Overbay explained the overall project, Book Embeddings of Graphs, this way:

“Our project involves drawing graphs (a collection of nodes with edges joining some of the nodes) on various objects so that no two edges cross. These graphs are used to represent networks. Two nodes that can directly communicate with each other are connected with an edge. If the edges represent wires in chip design, we may not want wires to cross. So we look for ways to draw the graphs so that edges don't cross (if possible), or we look for ways to create as few layers with non-crossing edges as possible.”

“The structure we use is called a book. Think of this as a line – the spine – that joins together a bunch of pages, half planes connected together at this line. We place the nodes on the line and the edges on the half planes so that no two edges intersect. We look to minimize the number of pages needed to do this.”

“We then looked at changing the pages from half planes to cylinder pages, torus pages and Mobius pages and determined how that would change the optimal number of pages needed for many types of graphs.”

Trees in the Wind

Pansegrau, who like his research partner Lynn graduated with a Mechanical Engineering degree in May, gave a rundown of their work with Michelle Ghrist, associate professor of mathematics.

“Basically, we are modeling trees’ mechanical motion due to an external wind force,” he wrote.

“We analytically solved a fourth-order partial differential equation that incorporates the tree’s natural resistance to bending (a property of the tree's material), a height- and time-dependent forcing function that simulates wind, and a damping term that increases the resistive force as the tree’s speed increases. In the absence of wind, this friction-like term causes the tree to eventually stop swaying.”

“Since the wind function is dependent on time and height, we can simulate gusts of winds and also varying magnitudes of force with respect to height.”

He acknowledges that applications currently are limited due to assumptions they’ve made, such as “that the tree has a constant density as a function of height, which does not reflect real-world scenarios.” Eliminating some of these assumptions is a future goal.

How did it all get started?

“I had been inspired to model the motion of trees after being enthralled by very tall trees in an aspen grove blowing in the wind near Aspen, Colorado, while on a hike with my son years ago,” Ghrist said. “This project proved to be an outstanding fit for Max, as the starting model is similar to that of vibrations of a mechanical beam, which he had studied in other classes.”

"The work will continue," Ghrist said.

“Future work on the project includes studying the effect of varying our parameters and incorporating different functions for modeling the wind,” she said. “We plan to publish our work in a journal.”

“We had originally hoped to also model the motion of animals’ tails, but that project will wait for future students,” referencing the math department’s long history of providing opportunities for undergraduates – both math and other STEM majors – to participate in research.

A New Tool for Modeling

Lombardi, from Issaquah, Washington, will be a senior in the fall as he pursues his Applied Mathematics major with an Economics concentration, and a Computer Science minor. He presented a poster on research he is doing with Ghrist.

“This project aims to develop adaptable methods of numerical approximation that modelers can directly tune to a given problem that they are working on,” he wrote. “This research hopes to give modelers more options for interesting problems than just the standard ‘out of the box’ methods normally used.”

He explained that the methods he is developing are mainly applied to problems in orbital mechanics such as predicting the orbits of satellites and asteroids.

The origins of the research came out of Ghrist’s work as an analyst at Air Force Space Command in 2010-11 as part of a sabbatical while she was a professor at the U.S. Air Force Academy.

“We start from the numerical methods underlying how the Air Force is approximating trajectories of objects in outer space and then try to find ways to improve these methods,” she said.

Initially, applied math major Matt Rhilinger also worked on the project. He subsequently graduated and is in graduate school at Colorado State University.

Lombardi plans to continue research into methods for numerical approximations this fall as part of his Senior Honors Project.

Before that, he’s one of eight College of Arts and Sciences students to receive a Morris Undergraduate Research Fellowship, established in 2017 by GU graduates Scott and Liz Morris. The fellowships support student-faculty teams doing original projects carried out during the summer. Both the student and the faculty mentor, in this case Ghrist, receive a stipend.

“I will be combining the research I featured at the conference with research that I am doing this summer into a definitive project on free-parameter multistep methods,” he said, adding that Ghrist “has been an invaluable mentor not only on this project but also my academic career.”

Students, Faculty Exchange Compliments

Like Martin and Lombardi, the other students were quick to heap praise on the math faculty who guided them.

“Dr. Ghrist deserves all of the credit in the world for inspiring her students and being the most determined and hard-working person I have ever met,” said Lynn, of Spokane Valley. ”We are lucky to have been able to work with her for so long on this project.” Ghrist encouraged her to add the Applied Mathematics minor to her engineering major, she said.

Pansegrau, of Morgan Hill, California, added his enthusiasm: “We definitely could not have been able to complete everything without her help and inspiration.” He completed a Psychology minor besides his engineering degree.

The faculty returned the applause.

“This group of three research students has been quite productive, motivated, clever, and insightful,” Overbay, speaking for herself and McKenzie, said of Linthacum, Tan and Martin. “It has been a delight working with them.”

“Sara Jane (Lynn) asks outstanding questions that showcase her natural ability to get to the heart of a matter,” Ghrist said. “Both she and Max (Pansegrau) are hard-working, conscientious students who have good mathematical instincts. I have greatly appreciated this opportunity to continue working with Max and Sara Jane and getting to know them better.”

Some of the same traits characterize Lombardi, Ghrist said.

“Ben has an outstanding attitude and work ethic,” she said. “He picks up new ideas and techniques quickly and is very well-read. I appreciate that Ben does not shy away from intellectual challenge. I am always intrigued to see what he has accomplished between our meetings as he works very independently.”

“It is rare to have the opportunity to work with a student for two years on a project, but that amount of time allows for a student to go so much further. As Ben and I start our third year of work together, I am excited to see where this project ends up going.”

And it bodes well for new Zags, the math professor added.

“I am so grateful that Ben and Matt joined me in tackling this new research area. Their work has opened up several new areas of research for future students. As a team, we have made great strides these past two years.”

To learn more about Mathematics Student Research.