Christopher Eur Adds Breadth to Faculty with Algebraic Geometry

Matroid theory is a way mathematicians describe the property of independence in a space, with uses across mathematics, physics, computer science and more.

Carnegie Mellon University Mathematician Christopher Eur takes a particular interest in matroids in his research, which focuses on algebraic geometry and its intersection with combinatorics, the study of counting of objects.

"I like thinking about how continuous and geometric objects interact with discrete and combinatorial objects," said Eur, who joined the Department of Mathematical Sciences as an assistant professor in September. "One may not think that these two have much interplay but, as it turns out, they do."

Eur was awarded research funding from the National Science Foundation (NSF) from 2023 to 2026 to further understanding of matroid theory (research grant DMS-2246518).

"This proposal delves deeper into just how much one can know or understand about discrete structures through geometry, and probes the boundary between the two," Eur said.

The project, called "Positive Vector Bundles in Combinatorics," is an example of applying algebraic geometry to understand combinatorial objects. The research seeks to understand objects like graphs and matchings through the geometric constructions called positive vector bundles.

"My proposal is that, with some of these geometric methods, we will have more ways and techniques to actually get our hands around them," Eur said.

The research may result in some concrete problems in discrete mathematics where the objective is to count certain quantities for graphs, vector configurations, polytopes or other applications.

Eur began taking a combinatorial view of algebraic geometry during his doctoral studies. It was then that his advisor David Eisenbud introduced him to the work of June Huh, a 2022 Fields Medalist, the highest scientific honor in mathematics.

"My research direction changed almost overnight when I looked at some of June's papers," Eur said. "Since then, I have worked with June, as well as other people in the field, and I've been having a lot of fun."

The open-access journal Forum of Mathematics, Pi will publish an article "Stellahedral geometry of matroids" that he co-authored with Huh and Matt Larson.

Eur was the Benjamin Peirce Postdoctoral Fellow at Harvard University from 2021 to 2024. This fellowship is awarded to scholars with significant promise in research and a strong teaching record. Eur also worked at Stanford University as a National Science Foundation postdoctoral fellow from 2020 to 2021 (DMS-2001854). He earned his Ph.D. in mathematics from the University of California, Berkeley and a bachelor's degree in mathematics from Harvard University.

"In general, I've been quite lucky to have many wonderful mentors and teachers," Eur said.

Eur, who enjoys hiking and climbing, is embarking on the next leg of his career in Pittsburgh.

"My hobbies have a large overlap with other mathematicians — if you ask a random mathematician, the chances are, they either hike or do rock climbing," he joked.

He said he looks forward to research and teaching at Carnegie Mellon.

"CMU has a strong tradition in discrete math, but there has not been as much focus in algebra or algebraic geometry, and so part of what I'm excited about is to bring that expertise here and interact with many other faculty," Eur said.

Eur currently teaches a course in graduate algebra this fall. He plans to develop a new course on algebraic geometry.

"The prospect for growth is exciting," Eur said. "We'll see what kind of journey I take!"