A professor dedicated to solving complex issues in higher mathematics has successfully addressed two major problems that have baffled mathematicians for years.
A Rutgers University-New Brunswick professor committed to unraveling the intricacies of higher mathematics has achieved solutions to two significant problems that have confounded mathematicians for many years.
The resolutions to these enduring problems could deepen our understanding of the symmetries found in natural structures and phenomena, as well as the long-term behavior of various random processes applicable across a range of disciplines including chemistry, physics, engineering, computer science, and economics.
Pham Tiep, the Joshua Barlaz Distinguished Professor of Mathematics in the Department of Mathematics at the Rutgers School of Arts and Science, has successfully proven the Height Zero Conjecture, established by Richard Brauer in 1955, a prominent German-American mathematician who passed away in 1977. This proof—widely regarded as one of the most considerable challenges in representation theory of finite groups—was documented in the September edition of the Annals of Mathematics.
“A conjecture is essentially a hypothesis that you believe holds some truth,” explained Tiep, who has pondered the Brauer issue for much of his career and has intensely focused on it for the last decade. “However, conjectures must be proven. I aimed to make progress in the field but never anticipated that I would resolve this one.”
In a way, Tiep and his team have been following a roadmap of challenges that Brauer detailed in various mathematical conjectures during the 1950s and 60s.
“Certain mathematicians possess an extraordinary intellect,” Tiep noted about Brauer. “It’s as if they originate from another planet or dimension. They have the ability to perceive hidden phenomena that elude others.”
In another significant achievement, Tiep tackled a challenging problem associated with Deligne-Lusztig theory, a crucial aspect of representation theory. This achievement relates to traces, which are a key feature in a matrix—a rectangular array of numbers. The trace of a matrix consists of the total of its diagonal elements. The findings are elaborated upon in two papers, the first published in Inventiones mathematicae, vol. 235 (2024), and the second in Annals, vol. 200 (2024).
“Tiep’s exceptional research and expertise regarding finite groups have helped Rutgers maintain its reputation as a leading global center in this area,” stated Stephen Miller, a Distinguished Professor and Chair of the Department of Mathematics. “One of the monumental achievements in 20th century mathematics was the classification of what are inaccurately termed ‘simple’ finite groups, which was led from Rutgers and where many fascinating examples were discovered. Through his remarkable contributions, Tiep enhances the international profile of our department.”
According to Tiep, the insights gained from these solutions are expected to significantly advance mathematicians’ knowledge of traces. Additionally, the findings could pave the way for breakthroughs concerning other significant mathematical problems, including those suggested by University of Florida mathematician John Thompson and Israeli mathematician Alexander Lubotzky.
Both discoveries contribute to the field of representation theory of finite groups, which is a part of algebra. Representation theory plays a vital role in various mathematical domains including number theory and algebraic geometry, and it also is relevant to the physical sciences, including particle physics. This theory utilizes mathematical entities known as groups to explore symmetries in molecules, facilitate message encryption, and create error-correcting codes.
By applying the principles of representation theory, mathematicians can convert abstract shapes found in Euclidean geometry—some incredibly complex—into numerical arrays. This is done by identifying specific points in each three-dimensional shape and mapping them to numbers arranged in rows and columns.
Moreover, Tiep emphasized that the reverse process must also be feasible: one should be able to reconstruct the original shape from the sequence of numbers.
In contrast to many of his peers in the physical sciences who often rely on sophisticated equipment for their research, Tiep utilizes just a pen and paper. His efforts have resulted in five books and over 200 publications in esteemed mathematical journals.
He sketches mathematical equations or notes that outline logical sequences. Additionally, he maintains ongoing discussions—either face-to-face or via Zoom—with colleagues as they meticulously work through proofs together.
However, Tiep also remarked that progress often emerges from introspection; ideas frequently come to him when he least anticipates them.
“Sometimes I might be strolling with my children, gardening with my wife, or simply busying myself in the kitchen,” he shared, adding, “My wife says she can always tell when I’m deep in thought about mathematics.”
In pursuing the first proof, Tiep collaborated with Gunter Malle from Technische Universität Kaiserslautern in Germany, Gabriel Navarro from Universitat de València in Spain, and Amanda Schaeffer Fry, a former graduate student of Tiep’s now at the University of Denver.
For the second significant finding, Tiep partnered with Robert Guralnick from the University of Southern California and Michael Larsen from Indiana University. In the initial of the two papers addressing the issues concerning traces, Tiep worked alongside Guralnick and Larsen. Tiep and Larsen co-authored the second paper.
“Tiep and his co-authors have achieved trace bounds that are arguably the best we could ever hope to achieve,” Miller noted. “It’s a well-developed subject that holds importance for many reasons, making progress challenging—but the applications are vast.”
