Aliakbar Daemi, an associate professor of mathematics, has been awarded more than $240,000 by the National Science Foundation (NSF) for a project called “Instantons, Knots, and Surgery.”

The award will support Daemi’s development of new mathematical tools to study knots and the structure of three- and four-dimensional spaces. Drawing on gauge theory, a mathematical framework rooted in physics, the project will study longstanding questions about how these spaces can be recognized and distinguished.

A central focus of the project is Dehn surgery, a technique for constructing and studying three-dimensional spaces. “Surgery in topology is somewhat analogous to surgery in medicine. We excise a neighborhood of a knot and glue it back in a different way,” Daemi said. “We are interested in understanding how these operations affect the resulting space and, in particular, whether different surgeries can produce the same space.” 

The project will also investigate how much information about a knot can be recovered from its surrounding space, using methods based on persistent homology. These methods track how mathematical structures change as parameters vary and also have applications in data analysis.

The award will support student training, research dissemination, and outreach activities, including a local elementary-school math club and a seminar on Mathematics and AI.

“I’m grateful for this award, which supports both my research and opportunities to involve students,” Daemi said. “Our recent work has opened up several promising directions, and I look forward to developing these ideas further. I hope they will lead to progress on longstanding problems about knots and three-dimensional spaces, while also providing new tools for addressing other questions in topology.”