You know geometry: circles, squares, maybe a polygon. You also know about dimensions: a line, a plane and a 3D volume. Now how about G2-geometry occurring in seven-dimensional space?
If it’s not ringing a bell, don’t feel bad. G2-geometry is not only a specialized field, but also a relatively new advance in a discipline going back thousands of years. The contemporary study of geometry has moved far beyond the foundations of what most people think of as classic geometry, established by the Greek mathematician Euclid.
Today, one of the frontiers of geometry is G2-geometry — and some of its leading faculty experts are right here at Duke, in the Department of Mathematics.
Leading faculty experts… and three undergraduate math majors: Matthew Chen, Jack Qian and Paul Rosu.
As part of a Math+ project on the computational exploration of geometric flows in G2-geometry, Chen, Qian and Rosu wrote code to numerically solve a geometric flow and will be co-authors on a forthcoming paper alongside the project’s leaders, geometers Ilyas Khan and Alec Payne.
Applications for Math+ 2025 are now open.
Math+ is an 8-week summer collaborative undergraduate research program run by the Duke Mathematics Department. It is open to all Duke undergraduate students, and students from all backgrounds and experiences are encouraged to apply.
In 2025, the program is offering six projects on a variety of topics in pure and applied mathematics, including one 10-week project jointly offered between Math+ and Data+.
Math+ will run from May 20 to July 11, 2025 and participants receive a stipend. The application deadline is February 15, 2025.
For information, visit the Math+ 2025 website.
In mathematics, a "geometric flow" is a type of equation that describes how a geometric object — in this case, a manifold — can evolve over space and time, essentially allowing the object to "flow" and redistribute its shape based on certain properties. In simple terms, a manifold is a deceptively complex shape. If you were to plop yourself on the surface of a manifold and look around you, you’d assume that you had landed on a flat surface, not unlike our experience on Earth. Zooming out though, someone flying overhead would see that the manifold’s overall shape can be anything: a sphere, a torus, and it can also change shape — like a block of clay. A geometric flow can explain, or guide, these changes in shape.
The advent of geometric flows is considered one of the largest advances in early 21st century mathematics. “Thirty years ago, people were stumped by some major open problems in geometry,” Payne said. “But now we’ve figured out a lot, because of geometric flows.”
The hope for the Math+ project team was to study G2-Laplacian flow, a geometric flow of seven-dimensional manifolds, which is something that Payne says had not been “studied enough” before this project.
That’s where undergraduate students Chen, Qian and Rosu come in.
Working in collaboration with Payne and Khan — who are both assistant research professors of Mathematics — and graduate student George Daccache, Chen, Qian and Rosu used MATLAB to simulate certain flows on manifolds and visualize “singularities,” which refer to points in time or space where the smooth evolution of the flow can no longer continue: the clay cracks.
“Think of it like trying to spread out heat,” Payne explains. “But the heat gets trapped somewhere and causes a problem and something breaks.”
Geometers study G2-Laplacian flow because it holds promise as a tool for solving major open problems about the geometry of seven-dimensional manifolds. But to use as a tool, they first have to understand its singularities. “Understanding the singularities of the flow is the primary hurdle to using the flow to solve these open problems,” Payne said.
The question for the team was whether any singularities could form along the flow of their seven-dimensional manifolds.
“Can we prove that a singularity can form along this flow?” Payne asked. “We knew that it was theoretically something that could be solved by Duke students who were sufficiently good at coding and numerical simulations, and that’s basically what happened. These students uncovered some amazing behavior of these geometries in seven dimensions that we didn't expect.”
Working on a theoretical math project like this one requires an enormous amount of background knowledge, and Payne and Khan spent weeks providing the three undergraduate students with intensive trainings and lectures so they could tackle the problem as true collaborators.
“At the start of the project, I was like — why are we having so many lectures?” Chen said. “But as the days passed, I came to understand how important it was to understand the project as a whole, in order for us to have the language to collaborate.”
“They didn’t have to provide us with as much background as they did for the code to get written, but that would have divorced us from knowing exactly what we were doing,” Rosu said. “Instead, they took a lot of time and effort to work with us so we could learn the background first, which then allowed us to actually own the project.”
The collaborative nature of the project — and the mentorship and teaching — has influenced the undergrads’ interest in geometry and future academic pursuits.
“I was expecting to get a taste of geometry at Duke at some point, but, after this project, I’m much more serious about exploring the field,” Rosu said.
