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UD mathematicians awarded $275K for ‘pure’ math research, student mentoring

By Dave Larsen

The National Science Foundation awarded two University of Dayton mathematicians nearly $275,000 to research the building blocks of biological systems and the geometry of four-dimensional spaces.

Both projects involve “pure” mathematics, which explore abstract concepts and theoretical frameworks. However, they could contribute to biology, physics and engineering applications.

“Areas that were considered at one time too abstract have been exactly what is needed for modern problems,” said Alan Veliz-Cuba, associate professor in the UD Department of Mathematics. “The pure math of today becomes the applied math of tomorrow and can be tools needed for complex, real-world problems.”

Veliz-Cuba received a three-year, $116,354 grant to research a mathematical theory for the biological concept of “modularity,” in which complicated structures are generated from simpler building blocks, or modules.

His award is part of a larger $670,000 NSF grant that includes researchers at Iowa State University, the University of Florida and University of Kentucky, which include experimentalists working with plants and vertebrates.

Jun Li, assistant professor in the UD Department of Mathematics, received a two-year, $156,571 grant to explore four-dimensional “symplectic” structures by leveraging insights from two-dimensional models. In mathematics, symplectic refers to a type of geometric structure on a smooth even-dimensional space formed by the relationships between lines.

Li’s grant is part of the NSF’s Launching Early-Career Academic Pathways in the Mathematical and Physical Sciences (LEAPS: MPS) program, which supports research of pre-tenure faculty in the mathematical and physical sciences, with an emphasis on minority-serving or predominantly undergraduate institutions.

Li's grant will fund up to three undergraduate students to work on research projects.

“In addition to advancing fundamental knowledge, the grant will foster collaborative opportunities for students at the University of Dayton and more broadly the Miami Valley region, providing them with exposure to high-level research at the intersection of math and physics,” Li said.

math grants, 2024

Veliz-Cuba’s grant includes funding for graduate, undergraduate and high school students at UD and the institutions of his research partners. He plans to hire one undergraduate student researcher.

“If they are a biology major, they could work on a small gene network,” Veliz-Cuba said. “If they are more math-oriented, they could do simulations. There is an opportunity for everyone — that is a good thing. There are questions from more the biological side and more the mathematical side, or a combination.”

Veliz-Cuba is working to create mathematics needed to understand, predict and control biological systems in terms of their building blocks. He and his collaborators are studying the modularity of specific gene regulatory networks underlying salamander limb regeneration as well as hormone regulation in plants.

Their goal is to come up with a system for quantifying biological modules.

“To be able to say, for example, this gene network in this plant, Arabidopsis, is in charge of something and that’s because it has these three, four, five modules and this is the way they get combined, and this is exactly how that combination gives rise to this behavior that we see,” he said.

Li’s project looks at the underlying mathematical structure of our space-time universe, which is called a four-dimensional manifold. However, little is known about the geometry of four-dimensional spaces.

“Initially formulated from the equations of Hamiltonian mechanics, symplectic structures have grown into an important abstract mathematical topic that is particularly powerful for studying four-dimensional spaces,” Li said. “This project focuses on understanding the topology and dynamics of four-dimensional manifolds with a symplectic structure, using cutting-edge tools to make novel advances.”

Topology studies the fundamental properties of space that remain unchanged even when the structure is continuously bent, while dynamics studies the self-mappings of space that preserve a given structure.

Since joining the UD faculty in fall 2022, Li has taught courses in set theory and topology, sparking interest among undergraduate students in these areas. This has led to projects and presentations at workshops and UD’s Bro. Joseph W. Stander Symposium, an annual showcase of student research.

“In the coming years, I hope to continue mentoring and supporting undergraduates, particularly those interested in topology, dynamics and their applications in physics,” Li said.

Photos: (top) Jun Li (left) and Alan Veliz-Cuba; (middle) Jun Li (left), assistant professor in the UD Department of Mathematics, and some of his students.

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