Professors Hadi Salmasian and Alistair Savage, from the Department of Mathematics and Statistics, first crossed paths over 20 years ago as graduate students at Yale University. Though they stayed in touch, it wasn’t until recently that they found the right opportunity to collaborate.
What brought them together was a shared fascination with a branch of mathematics called representation theory, which explores how abstract structures reveal hidden patterns and symmetries.
At the heart of their research lies the theory of quantum groups, a modern extension of classical mathematics used in geometry and physics. Quantum groups are algebraic systems that capture the hidden symmetries of quantum systems, like those studied in quantum mechanics.
“These structures are fascinating because they give us new ways to describe the universe,” says Salmasian. “Even though the definitions are abstract, their applications in physics and beyond are very real.”
To bring their different perspectives together, they co-supervised a postdoctoral fellow, Yaolong Shen. “Yaolong Shen was a catalyst,” says Salmasian. “He brought in fresh ideas and a strong technical background that helped push the project forward faster than we anticipated.”
Quantum groups: Symmetry in a new light
The team’s recent breakthrough focuses on iquantum groups, a cutting-edge area of the field. They used tools from category theory and string diagrams, visual methods that let mathematicians “draw” equations instead of writing them, to discover simpler, more intuitive ways to describe extremely complex algebraic structures. “Think of it as replacing pages of dense equations with a single picture,” says Savage. “It’s not just easier to understand — it opens up entirely new perspectives.”
Underpinning tomorrow’s technologies
While the work may feel far removed from everyday problems, its impact could be profound. Much like the mathematics of prime numbers became the foundation of modern cryptography, today’s abstract discoveries could underpin tomorrow’s technologies.
Quantum groups already play a role in areas of quantum computing, a field expected to revolutionize industries from health care to finance. The mathematical tools developed by Salmasian, Savage and Shen may help shape the algorithms and systems that make quantum computing a reality.
“Mathematics has always been the foundation for future discoveries,” says Salmasian. “The work we’re doing now may not have immediate applications, but it builds the theoretical base that science and engineering will stand on decades from now.”
Tackling more complex quantum groups
What began as a collaboration to learn from one another has blossomed into a research program with wide-ranging potential. The team is now expanding its methods to tackle more complex families of quantum groups, with an eye toward deeper application in both mathematics and physics.
For Salmasian, Savage and Shen, the project is more than just an academic exercise. It’s a reminder that even the most abstract branches of mathematics can reveal surprising beauty and serve as a guidepost for humanity’s future discoveries.
“Start with something simple, and patterns emerge,” says Salmasian. “That’s how mathematics works, and that’s how progress is made.”
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