What is the large-scale shape of the universe? If you traveled far enough and fast enough, would you come to where you started, like a sailor circumnavigating the Earth? Or would you keep moving off indefinitely? Questions such as this lay behind the study of mirror symmetry and related questions in mathematics. Mathematicians in the University of Hawaiʻi at Mānoa Department of Mathematics are helping to spearhead a new $1.5 million National Science Foundation (NSF) project that will explore cutting-edge links between algebra, geometry and physics to develop tools with wide-ranging applications.
Daniel Erman, a faculty member in the mathematics department, will lead UH Mānoa’s part of the project, which totals about $238,000 of the larger award. The project focuses on deepening understanding of how algebra (the study of equations and symbols) and geometry (the study of shapes and spaces) interact in unexpected ways, with potential implications for fields ranging from computer science and robotics to theoretical physics.
Recent discoveries have revealed surprising links between algebraic geometry—where shapes are defined by equations—and symplectic geometry, which underpins many ideas in modern physics and engineering. By combining approaches from multiple subfields, the researchers aim to develop new mathematical tools that can bridge these areas.
Training opportunities for researchers, students
The work will also provide training opportunities for early-career researchers and students, including the creation of workshops, virtual seminars and publicly available computational tools. At UH Mānoa, the grant will support graduate student research, fund a postdoctoral fellow from 2026 to 2028, and host a 2026 workshop which will bring researchers from all over the country and will provide many opportunities for UH graduate students and math majors to learn about recent advances.
“This project represents the best of mathematical collaboration,” Erman said. “By pooling expertise across universities, we can tackle problems that no one group could solve alone, while also training the next generation of mathematicians here in Hawaiʻi and beyond.”
The collaborative effort, supported through NSF’s Focused Research Groups in the Mathematical Sciences program, includes researchers from the University of Southern California, University of Minnesota, Auburn University, University of Oregon and the University of South Carolina.
The UH Mānoa Department of Mathematics, housed in the College of Natural Sciences, continues to build national partnerships through faculty research while offering students unique opportunities to engage in high-level discovery.
