Vietnamese mathematician makes dual discovery, helping US university maintain world leadership in algebra

"A professor at Rutgers University-New Brunswick, who has devoted his career to solving the mysteries of mathematics, has solved two separate fundamental problems that have puzzled mathematicians for decades , " Phys.org , a UK science news site, reported on October 9.

That professor is Pham Huu Tiep , a Vietnamese mathematician born in 1963, a former student of Chu Van An High School, Hanoi.

Professor Pham Huu Tiep.

Phys.org 's comments came after Professor Pham Huu Tiep published a new scientific paper in the September issue of Annals of Mathematics . The paper provided a solution to a nearly seven-decade-old mathematical problem , posed by the eminent German-American mathematician Richard Brauer in 1955.

" Solution to these long-standing problems could further advance 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 stochastic processes that arise in fields ranging from chemistry and engineering physics to computer science and economics ," writes Phys.org .

Untying a 70-year-old knot in mathematics

Brauer's problem is known as the "Pitch 0 Conjecture" , in which Brauer predicts that for any finite group G and p primes, some arithmetic properties of the more indivisible representations of the group G in a special section called the p-block B , must be controlled by defect groups (D).

In simple terms, this is a prediction by Brauer in a field of algebra called "group theory" , which is concerned with representing quantities in groups.

A group is a collection of objects that we can combine together according to a certain rule, for example, when you rotate a square by 90 degrees, 180 degrees, or 270 degrees. These rotations form a "group" because they can be combined together and follow certain rules.

When mathematicians study groups, they often want to represent them in a way that is easier to understand. One common way is to represent groups using matrices (a table of numbers). Representing groups means writing down the elements of a group (like ways to rotate a square) using matrices.

Brauer's "zero pitch conjecture" is about the complexity of the "pieces" in a group's representation. When we decompose a group into smaller and simpler parts (like breaking a puzzle into pieces), each piece has a pitch, which is a non-negative number.

This conjecture says that for certain groups, all of those pieces have height 0, meaning they are as simple as possible, but only if the group satisfies certain conditions.

The "zero pitch conjecture" was proposed by German-American mathematician Richard Brauer in 1955.

Until recently, Brauer's conjecture was just a guess. That is, Brauer thought it was true, and mathematicians had tested it in countless cases and found it to be true. But it was impossible to prove it true.

" Some mathematicians with rare intelligence [like Brauer] seem to come from another planet or another world. They have the ability to see hidden things that others cannot ," said Professor Pham Huu Tiep about how Brauer came up with his hypothesis in 1955.

'A hypothesis is an idea that you believe to some degree is true. But hypotheses have to be proven ,' he added.

In their new study published in the journal Annals of Mathematics , Professor Pham Huu Tiep and 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 at the University of Denver, have fully proven Brauer's zero pitch conjecture.

This success is considered to have untied an extremely important knot in group theory that has existed for the past 70 years.

Part of the solution of Professor Pham Huu Tiep in new research.

Helping American universities maintain world leadership in a field of algebra

But that 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 also successfully solved a difficult problem called the Deligne-Lusztig theory.

It 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 of linear algebra.

By clarifying the Deligne-Lusztig theory, Professor Pham Huu Tiep said his solution could provide insights that could help other mathematicians solve many other major problems in mathematics, including conjectures put forward by mathematician John Thompson of the University of Florida and Israeli mathematician Alexander Lubotzky.

Commenting on the two new studies by Professor Pham Huu Tiep, Stephen Miller, Professor Emeritus and Chair of the Department of Mathematics at Rutgers University-New Brunswick, said: "Professor Tiep's high-quality work and expertise on finite groups have helped Rutgers University maintain its position as a world-leading center in this field.

One of the great achievements of 20th century mathematics was the classification of "simple" finite groups , which are so called but perhaps misnamed [because they are actually very complicated]. The most exciting and pioneering discoveries in this field have been led by Rutgers University. Through his incredibly prolific career, Professor Tiep has given our Mathematics Department an international presence."

Through his incredibly prolific career, Professor Pham Huu Tiep has helped the Mathematics Department of Rutgers University-New Brunswick gain international visibility."

Both of Professor Pham Huu Tiep's breakthroughs were important 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 science, including particle physics.

Through mathematical objects called groups, representation theory has also been used to study symmetries in molecules, encode messages, and create error-correcting codes.

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

Professor Pham Huu Tiep said the reverse operation must also be effective. One needs to be able to reconstruct shapes from sequences of numbers. And to do that, group representation theories need to be developed.

"I hope to promote this field ," said Professor Pham Huu Tiep.

Study math with pen and paper

Professor Pham Huu Tiep was born in 1963 and is a former student of Chu Van An School, Hanoi. He participated in the International Mathematical Olympiad (IMO) held in England in 1979 and won a Silver Medal.

In 1980, he went to study Mathematics - Mechanics, Lomonosov State University, former Soviet Union. After graduating from university in 1985, he continued his research and defended his associate doctoral thesis (now called doctorate) in 1989, then his doctoral thesis (now called doctorate of science) in 1991.

In 1996, he moved to the United States and worked at many universities such as Ohio University, Florida University, and Arizona University. In 2013, he was elected an honorary member of the American Mathematical Society. From 2018 to present, Professor Pham Huu Tiep has worked at Rutgers University, and collaborated with the Mathematical Sciences Research Institute (MSRI) Berkeley, and the Institute for Advanced Study Princeton.

Throughout his mathematical career up to now, Professor Pham Huu Tiep has published 5 monographs and more than 200 scientific articles in the world's leading mathematical journals.

Professor Pham Huu Tiep in a lecture on Brauer's hypothesis.

Unlike many of his colleagues who often use complex devices to work, Professor Pham Huu Tiep said he only uses pen and paper to conduct research. He always writes down mathematical formulas or sentences representing logical chains, then engages in continuous discussions with colleagues, both in person and online via Zoom.

However, Professor Pham Huu Tiep said his discoveries often come when he least expects them. " It could be when I'm walking with my kids, or gardening with my wife, or fiddling around in the kitchen. My wife says she always knows when I'm thinking about math," he said.