Dr. Vu Khac Ky, a Mathematics lecturer at FPT University, and his collaborator have obtained a complete proof of the Courtade–Kumar conjecture, one of the most important open problems in information theory. The work demonstrates that a small research team in Vietnam can directly contribute to fundamental problems that are also being pursued by leading research groups worldwide.
Recently, Vahab Mirrokni, Vice President at Google Research and head of research teams in Algorithms and Optimization at Google, announced a proof of the Courtade–Kumar (CK) conjecture by a Google research team, while also acknowledging and congratulating an independent solution by the Vietnam-based research team of Dr. Vu Khac Ky of FPT University and Professor Tuan Tran of the University of Science and Technology of China.
In announcing the result, Mirrokni described Courtade–Kumar as a “long-standing central open problem at the intersection of information theory and the analysis of Boolean functions.” He also noted that the monograph by Yu and Tan describes the conjecture as “one of the most important open problems in information theory.”
According to Mirrokni, the significance of CK extends beyond the problem of maximizing mutual information. The conjecture is connected to a broad family of extremal problems, functional inequalities, and isoperimetric inequalities on Boolean spaces, including connections to the inequalities of Talagrand and Bobkov and, in certain regimes, Harper’s classical edge-isoperimetric inequality.
Remarkably, at almost the same time, two completely independent research teams arrived at a solution through very different approaches.
Research Teams from Vietnam and Google Independently Reach the Same Result
The Courtade–Kumar conjecture, also known as the Most Informative Boolean Function Conjecture, was proposed by Thomas Courtade and Gowtham Kumar in 2013. The problem arises from a natural question in communication and information processing: when data becomes corrupted by noise during transmission, how should the original information be selected and processed so that as much information as possible can ultimately be preserved?
Surprisingly, the Courtade–Kumar conjecture proposes that the optimal strategy is extremely simple: instead of combining multiple bits according to a complex rule, it is sufficient to retain just one bit. Despite this intuitive formulation, for more than a decade, researchers around the world had only been able to solve individual cases of the conjecture.
About ten years ago, while conducting postdoctoral research with the Information Theory group at the Chinese University of Hong Kong (CUHK), Dr. Ky first learned about the conjecture from Professor Chandra Nair. He pursued the problem seriously for about two years without success. After joining FPT University (FPTU) as a lecturer in January 2019, he continued to revisit the problem from time to time, but repeatedly had to set it aside because he had not found a mechanism powerful enough to overcome the known barriers.
In 2025, Dr. Ky returned to the problem more seriously while researching information geometry and exploring the use of geometric tools to approach questions in machine learning. He realized that the Courtade–Kumar conjecture also contains structures naturally related to entropy, mutual information, and how information changes under noise. This prompted him to approach the problem again from this perspective.
During the same period, he began collaborating with Professor Tuan Tran, currently a specially appointed professor at the University of Science and Technology of China and a leading expert in discrete probability and combinatorics. Professor Tran’s tools from combinatorics, probability, and discrete structures complemented Dr. Ky’s entropy- and analysis-based approach, allowing the two researchers to continuously test, eliminate, and refine their ideas.
“AI does not replace the mathematical proof itself, but it helps us determine very quickly whether an idea is worth pursuing. When an approach is wrong, finding a counterexample early can save weeks of work. When a structure appears to be correct, we can focus on understanding why it works and turning it into a rigorous argument,” Dr. Ky said.
Through this process of experimentation and elimination, a proof structure gradually emerged, combining entropy estimates, the Fourier spectrum of Boolean functions, and several sharp inequalities. The researchers progressively expanded the range they could control, resolved the remaining cases, and ultimately arrived at a proof for the general case.
“The greatest difficulty was the enormous gap between the statement of the problem and the proof. The statement takes only a few lines to understand, but the most natural methods usually solve only a particular parameter range or a special class of functions. 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,” Dr. Ky explained.
As their work progressed, several new structures emerged, particularly the combination of entropy estimates, the Fourier spectrum of Boolean functions, and sharp inequalities. From these components, the researchers gradually expanded the range over which the conjecture could be established, addressed the remaining cases, and ultimately obtained a proof for the general case.
The paper “Dictators Are Most Informative,” by Vu Khac Ky and Tuan Tran, was posted on arXiv, the online repository for scientific preprints, on September 21, 2026. The work proves the conjecture for all Boolean functions, including the unbalanced case. The story behind the publication is equally notable.
Before releasing the paper, Dr. Ky contacted Professor Chandra Nair by email as a gesture of respect, informing him that he and Professor Tran had found a solution. Professor Nair replied that his own group, together with researchers from Google and several universities, had also just completed a proof of the same problem.
The two teams subsequently exchanged their findings and discovered that they had independently reached the same result using substantially different methods. The manuscript by the seven-member international team is more than 250 pages long, while the paper by the two researchers from Vietnam is approximately 36 pages. However, the difference in length does not indicate that one solution is superior to the other. Each approach has distinct strengths that will need to be assessed by the mathematical community over time.
After learning of each other’s results, and because Professor Nair’s team needed additional time to finalize a lengthy work involving extensive calculations, Dr. Ky and Professor Tran decided to delay the release of their paper so that the two independent solutions could be announced at the same time.
“For me, this is also a beautiful story about scientific research,” Dr. Ky shared.
The work by Dr. Ky and Professor Tran not only provides a solution to a long-standing open problem in information theory and Boolean analysis, but may also have broader implications. The tools developed in the proof, particularly the entropy inequalities and the approach to exploiting the structure of noise operators on Boolean spaces, may prove useful for other problems.
The achievement also demonstrates that a small research team in Vietnam can directly contribute to fundamental problems being pursued by leading research groups worldwide, provided the right problem is chosen and pursued with sufficient persistence.
An Environment Open Enough to Explore New Ideas and Respect Academic Independence
Before becoming a Mathematics lecturer at FPTU, Dr. Vu Khac Ky graduated from Hanoi University of Science through the university’s Talented Bachelor Program in 2009. He then pursued a master’s degree at TU Kaiserslautern in Germany. In 2012, he was the only student in France to receive a doctoral research scholarship from Microsoft Research Cambridge.
After earning his PhD from École Polytechnique in France, he spent two years working at the ITCSC research institute in Hong Kong before returning to Vietnam. He was also a core member of Flyspeck, one of the largest projects in formal mathematics, which sought to construct a computer-verified proof of the Kepler conjecture, a problem that had remained unresolved for nearly 400 years.
Looking back on his research career, Dr. Ky said that before the Courtade–Kumar work, one of the most important periods for him was his doctoral study at École Polytechnique under the supervision of Professor Leo Liberti. Rather than limiting doctoral students to a narrowly defined topic, Professor Liberti encouraged him to explore new directions and approach problems from multiple disciplines.
This gave Dr. Ky opportunities to work across areas ranging from optimization and compressed sensing to high-dimensional probability and machine learning theory. According to Dr. Ky, these experiences gradually shaped an interdisciplinary approach to research: rather than being constrained by disciplinary boundaries, he seeks to apply tools and perspectives from one field to problems in another.
“Some of the research directions from that period and the years that followed led to papers in Forum of Mathematics, Pi, Mathematical Programming, Mathematics of Operations Research, and Discrete & Computational Geometry. But what mattered more to me was developing the habit of always trying to view a problem through different mathematical languages. Sometimes, a tool learned in one field becomes the key to solving a seemingly unrelated problem,” Dr. Ky shared.
Professor Liberti’s open mentoring style and deep intuitive thinking have also influenced how Dr. Ky works with students at FPTU. He encourages students to explore different directions and identify questions genuinely worth pursuing rather than confining themselves to a fixed research path from the outset.
In teaching, Dr. Ky also considers helping students understand the structures and ideas behind a formula more important than simply teaching them how to solve a particular type of problem. He teaches courses including Discrete Mathematics, Linear Algebra, and Algorithms, and often seeks to show students that mathematics and computer science are not finished systems. Many questions can be stated very simply, yet no one knows the answer.
When conducting research with students, he encourages them to read broadly, experiment with different tools, and not be afraid to venture into other fields. He believes one of the most important things a mentor can do is create an environment open enough for students to explore new ideas, even when most of those ideas ultimately do not work. This is also what he received from his own mentor.
FPTU students supervised by Dr. Ky have gone on to pursue master’s and doctoral studies at leading universities worldwide. Nhat Mai was admitted to a master’s program at the University of Bonn in Germany; Vuong Tuan is currently a doctoral researcher at the University of Luxembourg; and Luong Chung was admitted to a master’s program at the University of Montreal in Canada.
Speaking about his current working environment, Dr. Ky particularly values FPTU’s respect for the academic independence of faculty members and the freedom lecturers are given to choose their research directions. For fundamental problems such as the Courtade–Kumar conjecture, it is difficult to set a timetable such as “a result must be produced within three months,” because researchers may work for a long time without knowing whether an approach will ultimately succeed.
“Being able to work in an environment that accepts that uncertainty is very important,” he emphasized.
Dr. Ky also appreciates that FPTU does not impose publication pressure based on quantity or require researchers to produce results within overly short timeframes. High-quality research, particularly fundamental research, sometimes requires time for an idea to mature. Applying pressure too early can easily lead to smaller, short-term results rather than more difficult questions with lasting value.
FPTU also enables him to maintain both teaching and research activities while encouraging international collaboration. The university’s recent recognition of this achievement has also been a significant source of motivation for him personally.
“In the long run, I think what matters even more than any individual paper is building an environment where lecturers and students feel respected, have enough freedom to pursue difficult questions, and are not overly afraid of failure. If results like this can build greater confidence and encourage more young people to pursue fundamental research, that would perhaps be the outcome I value most,” Dr. Vu Khac Ky shared.