In a double breakthrough, a mathematician helps solve two long-standing problems

A Rutgers University New Brunswick professor who has devoted his career to solving the mysteries of higher mathematics has solved two distinct fundamental problems that have intrigued mathematicians for decades.

Solutions to these long-standing problems could further improve our understanding of the symmetries of structures and objects in nature and science, as well as the long-term behavior of various random processes occurring in fields ranging from chemistry and science. physics to 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, produced a proof of the 1955 zero-height conjecture posed by Richard Brauer, a prominent German-American mathematician who died in 1977.

The proof of the conjecture – commonly considered one of the most remarkable challenges in a mathematical field known as finite group representation theory – is published in the Annals of Mathematics.

“A conjecture is an idea that you think has some validity,” said Tiep, who has thought about Brauer’s problem for most of his career and worked on it intensively over the past decade. “But conjectures must be proven. I was hoping to advance the field. I never expected to be able to solve this one.”

In a sense, Tiep and his colleagues followed a pattern of challenges that Brauer had presented to them in a series of mathematical conjectures posed and published in the 1950s and 1960s.

“Some mathematicians have this rare intelligence,” Tiep said of Brauer. “It’s as if they come from another planet or another world. They are able to see hidden phenomena that others cannot see.”

In the second step, Tiep solved a difficult problem in the so-called Deligne-Lusztig theory, which is part of the fundamental mechanism of representation theory. The breakthrough touches on traces, an important feature of a rectangular array called a matrix. The trace of a matrix is ​​the sum of its diagonal elements. The work is detailed in two articles. One was published in Mathematical inventionsthe second in Annals of Mathematics.

“Tiep’s high-quality work and expertise on finite groups has enabled Rutgers to maintain its status as a leading global center in this field,” said Stephen Miller, professor emeritus and chair of the Department of Mathematics.

“One of the great mathematical achievements of the 20th century was the classification of what are called, but perhaps incorrectly, “simple” finite groups, and it is synonymous with Rutgers – it was carried out from from here and many of the most interesting examples have been discovered here. Thanks to his incredible work, Tiep brings international visibility to our department.

The lessons learned from this solution will likely significantly improve mathematicians’ understanding of traces, Tiep said. The solution also provides insights that could lead to breakthroughs in other important mathematical problems, including conjectures posed by University of Florida mathematician John Thompson and Israeli mathematician Alexander Lubotzky, he added.

These two breakthroughs represent advances in the field of representation theory of finite groups, a subset of algebra. Representation theory is an important tool in many mathematical fields, including number theory and algebraic geometry, as well as in the physical sciences, including particle physics. Through mathematical objects called groups, representation theory has also been used to study the symmetry of molecules, encrypt messages, and produce error-correcting codes.

Following the principles of representation theory, mathematicians take abstract shapes that exist in Euclidean geometry – some extremely complex – and transform them into arrays of numbers. This can be achieved by identifying certain points that exist in each three-dimensional or higher shape and converting them into numbers placed in rows and columns.

The reverse operation should also work, Tiep said. You must be able to reconstruct the shape from the sequence of numbers.

Unlike many of his colleagues in the physical sciences who often use complex apparatus to advance their work, Tiep said he uses only pen and paper to conduct his research, which has so far resulted in five books and to more than 200 articles in leading mathematical journals. .

He writes down mathematical formulas or sentences indicating logical chains. He also engages in ongoing conversations – in person or on Zoom – with his colleagues as they progress step by step through an ordeal.

But progress can come from internal reflection, Tiep said, and ideas arise when he least expects them.

“Maybe I’m walking with our kids, gardening with my wife or just doing something in the kitchen,” he said. “My wife says she always knows when I think about math.”

On the first proof, Tiep collaborated with Gunter Malle of the Technische Universität Kaiserslautern in Germany, Gabriel Navarro of the Universitat de València in Spain, and Amanda Schaeffer Fry, a former graduate student of Tiep’s who is now at the University of Denver.

For the second advance, Tiep worked with Robert Guralnick of the University of Southern California and Michael Larsen of Indiana University. On the first of two papers that tackle and solve mathematical problems about traces, Tiep worked with Guralnick and Larsen. Tiep and Larsen are co-authors of the second paper.

“Tiep and his co-authors got bounds on the traces that are about as good as we could hope to get,” Miller said. “It’s a mature topic that is important in many ways, so progress is difficult and applications are numerous.”