Auburn mathematician wins NSF award to explore the hidden math of independence

Zach Walsh, Ph.D., assistant professor in Auburn’s Department of Mathematics and Statistics, has received a National Science Foundation award of $284,812 to support research on a fundamental question in mathematics, how to understand and measure “independence” in complex systems. 

In mathematics, independence has a technical meaning. Imagine a set of objects such as arrows pointing in different directions, pieces of a network or connections in a computer chip. Some subsets of these objects are independent, meaning that each element provides unique information, while others are dependent, meaning that some elements can be predicted from the rest. Mathematicians study this concept through structures called matroids, which capture the rules of independence in settings ranging from vector spaces in physics to networks in computer science. 

“Matroids give us a universal language for independence,” Walsh said. “They can describe relationships in vector spaces, networks and many other systems, all under the same set of rules.” 

Walsh’s project focuses on a special family called quadratically dense matroids. The name refers to how their size grows in relation to their rank, or complexity. Instead of growing slowly (linearly) or explosively (exponentially), they grow at a steady, quadratic rate. This middle ground is mathematically rich and largely unexplored. 

One goal of the project is to refine a major result in the field known as the growth rate theorem, which classifies how large matroids can get in certain categories. Walsh is especially interested in understanding the “quadratic” category more deeply, revealing new connections with other areas of mathematics. 

A surprising part of his work involves networks with edges labeled by elements of a finite group, an abstract algebraic structure that appears in many areas of math. Walsh suspects that in many important cases, the densest quadratic matroids come from these group-labeled networks. 

While the research is highly theoretical, it has potential applications in optimization, structural rigidity theory and network flow theory. “Minimizing or maximizing something subject to constraints is a common problem, and matroids have been used to study these problems before,” Walsh said. “I think the techniques we’re developing could be applied in new ways.” 

The NSF funding will support three years of work, including summer research opportunities for graduate students, travel for collaboration and bringing experts to Auburn. Walsh said more than half of the budget is dedicated to student research support. “It gives them time away from teaching duties to focus completely on advancing the project,” he said. 

In addition to the research, Walsh will connect Auburn’s discrete mathematics group with the community. He plans to engage local students through outreach programs such as Destination STEM for middle schoolers, Getting Under the Surface for elementary students and families, and Mathematical Puzzle Programs for high school students and teachers. 

“When I applied to grad school, I didn’t even know what a matroid was,” Walsh said. “I got paired with a supervisor who suggested I give them a try, and I’ve been working on them ever since.”