Vietnamese mathematicians solve decade-old information theory problem

Ky, a mathematics lecturer at FPT University in Vietnam, and Tran, a professor at the University of Science and Technology of China, posted their 36-page paper on arXiv, an online repository for scientific preprints, on Sept. 21, FPT University announced last week.

Proposed by mathematicians Thomas Courtade and Gowtham Kumar in 2013, the conjecture addresses a fundamental question in information theory: when data is corrupted by noise during transmission, how can as much information as possible be preserved?

Its proposed answer is surprisingly simple. Instead of combining multiple pieces of binary information, or bits, using a complex rule, retaining just one original bit provides the most information about the noisy data. Despite the simple premise, researchers had been unable to prove the conjecture in full for more than a decade.

At around the same time as Ky and Tran, an independent seven-member team involving researchers from Google and several universities also posted a proof on arXiv. The team includes Vahab Mirrokni, vice president at Google Research, and researchers from the Chinese University of Hong Kong, the University of Southern California and Carnegie Mellon University. Its paper, more than 250 pages long, presents a computer-assisted proof that it says resolves the conjecture in full generality.

In a post on X, Mirrokni described Courtade–Kumar as a "longstanding central open problem at the intersection of information theory and Boolean function analysis." He said his team had learned that Ky and Tran had independently obtained a different proof and congratulated them on the achievement.

Vietnamese mathematicians Vu Khac Ky. Photo courtesy of FPT

Ky first learned about the conjecture about a decade ago while conducting postdoctoral research with the Information Theory group at the Chinese University of Hong Kong.

He worked on the problem for about two years without success. After joining FPT University in January 2019, he occasionally returned to it but repeatedly set it aside because he could not find a way past existing barriers.

In 2025, he took up the problem again while researching information geometry and exploring how geometric tools could be applied to machine learning. He found that the conjecture had structures closely related to entropy, mutual information and how information changes when exposed to noise.

Around the same time, he began collaborating with Tran, a specialist in discrete probability and combinatorics. Tran's expertise complemented Ky's work in entropy and mathematical analysis, allowing them to test and refine different approaches.

Ky told FPT University that AI helped him and Tran quickly test ideas and identify counterexamples, allowing them to discard unsuccessful approaches and focus on promising ones.

He said the main challenge was that while the conjecture could be stated in a few lines, conventional methods could prove it only in limited cases.

"There were times when we thought we were nearly finished, only to discover a gap and have to start almost from scratch. So if I had to choose the single most important factor in this process, I would say it was the ability to persist with a problem after many failures."

Before publishing, Ky contacted Chandra Nair, a Chinese University of Hong Kong professor who had introduced him to the conjecture about a decade earlier. Nair told him his team, working with Google and other researchers, had independently reached the same result using a different method.

The teams exchanged findings and coordinated the release of their papers. "For me, this is also a beautiful story about scientific research," Ky said.

Ky graduated from the Talented Bachelor Program at Hanoi University of Science in 2009 before earning a master's degree at TU Kaiserslautern in Germany and a Ph.D. from École Polytechnique in France.

He later spent two years at the Institute of Theoretical Computer Science and Communications in Hong Kong before returning to Vietnam.

He was also a core member of Flyspeck, a major formal mathematics project that developed a computer-verified proof of the Kepler conjecture, a mathematical problem dating back nearly 400 years.