Brown receives NSF grant to connect algebra, geometry and physics

Michael Brown, associate professor in the Department of Mathematics and Statistics, has received a $193,584 grant from the National Science Foundation to support a three-year collaborative research project that bridges distinct areas of mathematics and physics.

The project, "Collaborative Research: FRGMS: Multigraded Commutative Algebra, Toric Varieties, and Homological Mirror Symmetry," brings together researchers from six universities to explore surprising links between commutative algebra and symplectic geometry.

Brown is the principal investigator and will collaborate with faculty at the University of Minnesota, University of Hawaii, Dartmouth, the University of Southern California and the University of South Carolina.

“The idea is to build a bridge between two fields that, at least on the surface, look very different,” Brown said. “We’re getting groups of people talking who hadn’t really interacted before.”

A key concept at the heart of this work is homological mirror symmetry, a phenomenon first discovered in the 1990s when physicists found solutions to mathematical problems that had long eluded researchers. Their methods sparked a new area of study that links algebraic and symplectic geometry in unexpected ways.

“To give an idea of how non-obvious this is, it’s like comparing two movies that seem completely unrelated,” Brown said. “But if you think of the color in one movie as the sound in the other, you realize they’re telling the same story. That’s what’s going on with mirror symmetry.”

In addition to advancing fundamental research, the grant will support a postdoctoral researcher at Auburn for two years.

“Postdocs inject new expertise and energy into a department,” Brown said. “It’s going to be great for everyone here, students, faculty and the department as a whole.”

Funding will also support travel for the team to collaborate across institutions and help organize a national conference. Brown emphasized that bringing together diverse perspectives is essential to sparking innovation.

“These are a lot of new ideas,” he said. “Part of the point is just to get people in the same room talking.”

The project includes mentoring and training opportunities for graduate students and early-career researchers. Auburn students will gain exposure to emerging areas of mathematical research and benefit from collaborations with scholars in fields not currently represented at Auburn.

“This area has huge potential for growth and impact,” Brown said. “Graduate students will have a chance to be part of that from the ground up.”

The project has already resulted in open-source computational tools that extend existing software capabilities. Brown said these tools make it possible to compute things that weren’t feasible before and are freely available for anyone to use.

“This research helps develop software at the interface of physics and mathematics,” he said. In the long term, Brown hopes the most lasting impact will be the new connections formed between traditionally separate fields of study.

“If you can really get people talking who haven’t talked before, that’s when new ideas happen,” he said. “That’s what leads to breakthroughs.”