“A professor at Rutgers University-New Brunswick, who has dedicated his entire career to solving the mysteries of mathematics, has resolved two separate fundamental problems that have puzzled mathematicians for decades,” according to Phys.org, a UK science news website reporting on October 9.
The professor is Pham Huu Tiep, a Vietnamese mathematician born in 1963, who was a former student at Chu Van An High School in Hanoi.
Professor Pham Huu Tiep.
The comment from Phys.org came after Professor Pham Huu Tiep published a new scientific paper in the September issue of the journal Annals of Mathematics. This paper provided a solution to a mathematical problem that has existed for nearly 70 years, proposed by the eminent German-American mathematician Richard Brauer in 1955.
“The solution to these long-standing problems could further enhance our understanding of the symmetry of structures and objects found in nature and science, as well as our understanding of the long-term behavior of many random processes arising in fields from chemistry and engineering physics to computer science and economics,” Phys.org wrote.
Unraveling a 70-Year-Old Knot in Mathematics
The problem posed by Brauer is known as the “Height Zero Hypothesis”, in which Brauer predicted that for any finite group G and any prime number p, certain arithmetic properties of the irreducible representations of the group G within a special part called the p-block B must be controlled by defect groups (D).
Simply put, this is a prediction by Brauer in a field known as “group theory” in algebra, concerning the representation of quantities by groups.
A group is a set of objects that can be combined according to certain rules; for example, when you rotate a square 90 degrees, 180 degrees, or 270 degrees. These rotations form a “group” because they can be combined and follow certain rules.
When mathematicians study groups, they often want to represent these groups in a more understandable way. A common method is to use matrices (a table of numbers) to represent the group. The representation of a group means expressing the elements of the group (such as the ways to rotate a square) using matrices.
The “Height Zero Hypothesis” by Brauer addresses the complexity of the “pieces” in the group representation. When we analyze a group into smaller and simpler parts (like dividing a jigsaw puzzle into pieces), each piece has a height, which is a non-negative number.
This hypothesis states that for certain groups, all those small pieces have a height of zero, meaning they are as simple as possible, but only if that group satisfies certain conditions.
The “Height Zero Hypothesis” was proposed by the German-American mathematician Richard Brauer in 1955.
Until recently, Brauer’s hypothesis was merely a conjecture. This means Brauer believed it to be true, and mathematicians tested countless cases and found it to be correct. However, proving this hypothesis true was impossible.
“Some mathematicians of rare intellect [like Brauer] seem as if they come from another planet or another world. They have the ability to see hidden things that others cannot,” Professor Pham Huu Tiep said about how Brauer proposed his hypothesis in 1955.
“A hypothesis is an idea that you believe to be true to some extent. However, hypotheses must be proven,” he added.
In his recent research published in the Annals of Mathematics, Professor Pham Huu Tiep and his colleagues, including Gunter Malle from the Technical University of Kaiserslautern in Germany, Gabriel Navarro from the University of Valencia in Spain, and Amanda Schaeffer Fry, his former graduate student now working at the University of Denver, have completely proven Brauer’s Height Zero Hypothesis.
This achievement is considered to have unraveled an extremely important knot in group theory that has persisted for the past 70 years.
A part of Professor Pham Huu Tiep’s solution in the new research.
Helping American Universities Maintain World-Leading Status in Algebra
But this is not the only remarkable achievement of Professor Pham Huu Tiep. Two months ago, he also published another study in the July issue of Annals of Mathematics. In it, Professor Pham Huu Tiep successfully solved a difficult problem known as Deligne-Lusztig theory.
This is also a problem in group representation theory. This breakthrough involves the trace of a matrix. The trace of a matrix is the sum of its diagonal elements and is an important concept in linear algebra.
By clarifying Deligne-Lusztig theory, Professor Pham Huu Tiep stated that his solution could provide deep insights, helping other mathematicians solve many other major problems in mathematics, including conjectures put forth by mathematician John Thompson from the University of Florida and Israeli mathematician Alexander Lubotzky.
Commenting on Professor Pham Huu Tiep’s two recent studies, Stephen Miller, a distinguished professor and Chair of the Mathematics Department at Rutgers University-New Brunswick, said: “The high-quality work and expertise of Professor Tiep in finite groups have helped Rutgers University maintain its status as a world-leading center in this field.
One of the great achievements of 20th-century mathematics is the classification of “simple” finite groups, which are so named but perhaps misnamed [because they are actually very complex]. The most interesting and pioneering discoveries in this field have been led by Rutgers University. Through his remarkable and vast career, Professor Tiep has elevated our Mathematics Department’s international presence.”
Through his remarkable and vast career, Professor Pham Huu Tiep has helped Rutgers University’s Mathematics Department gain international recognition.
Both breakthroughs by Professor Pham Huu Tiep are significant advances in the field of representation theory of finite groups, a subset of algebra. Representation theory is an important tool in many areas of mathematics, including number theory and algebraic geometry, as well as in physical sciences, including particle physics.
Through mathematical objects known as groups, representation theory has also been used to study symmetry in molecules, encode messages, and create error-correcting codes.
According to the principles of representation theory, mathematicians take abstract shapes existing in Euclidean geometry—some of which are extremely complex—and transform them into numerical matrices. This can be achieved by identifying certain points existing in each three-dimensional shape and converting them into numbers arranged in the rows and columns of the matrix.
Professor Pham Huu Tiep stated that the reverse operation must also be effective. There must be a capability to recreate the shape from the numeric sequences. And to achieve this, group representation theories need to be developed.
“I hope to promote this field,” Professor Pham Huu Tiep said.
Studying Mathematics with Pen and Paper
Professor Pham Huu Tiep was born in 1963 and is a former student of Chu Van An High School in Hanoi. He participated in the International Mathematical Olympiad (IMO) held in England in 1979, where he won a Silver Medal.
In 1980, he went to study Mathematics and Mechanics at Lomonosov Moscow State University, formerly the Soviet Union. Graduating in 1985, he continued as a research student, defending his Ph.D. thesis (now called a Doctorate) in 1989 and his doctoral thesis (now called a Doctor of Science) in 1991.
In 1996, he moved to the United States and worked at several universities, including Ohio University, the University of Florida, and the University of Arizona. In 2013, he was elected as a fellow of the American Mathematical Society. Since 2018, Professor Pham Huu Tiep has been working at Rutgers University and collaborating with the Mathematical Sciences Research Institute (MSRI) in Berkeley and the Institute for Advanced Study in Princeton.
Throughout his mathematical career to date, Professor Pham Huu Tiep has published five monographs and over 200 research articles in leading mathematics journals worldwide.
Professor Pham Huu Tiep lecturing on Brauer’s hypothesis.
Unlike many of his colleagues who often use complex devices for their work, Professor Pham Huu Tiep stated that he only uses a pen and paper to conduct his research. He always writes down mathematical formulas or logical sequences, and then engages in ongoing discussions with colleagues, both in person and online via Zoom.
However, Professor Pham Huu Tiep mentioned that his discoveries often arise at the moments he least expects. “It could be when I’m taking a walk with my children, gardening with my wife, or fiddling around in the kitchen. My wife says she always knows when I’m thinking about math,” he said.
