Recently, the entire nation has celebrated the achievement of Professor Ngo Bao Chau – a Vietnamese mathematician who has been awarded the most prestigious mathematics prize in the world, the Fields Medal. Many people want to understand the content of his work, but this is not an easy task. We can only provide readers with some brief and easily understandable explanations about this issue.
>>> Professor Ngo Bao Chau wins the Fields Medal in Mathematics
>>> Over 3000 people attended the welcoming ceremony for Professor Ngo Bao Chau
>>> Childhood of Ngo Bao Chau
In mathematics, a lemma is a hypothesis that has been proven or is certain to be proven, serving as a foundation for mathematicians to continue their research and achieve higher results.
Professor Ngo Bao Chau
In summary, without delving into complex mathematical concepts: In 1967, Canadian mathematician Robert Langlands proposed a series of bold hypotheses, most of which remain unproven to this day and will be the subject of research for many generations of mathematicians in the future. If proven, these hypotheses would unify many fields of modern mathematics, such as algebraic geometry and number theory.
One of the tools developed from the Langlands program is the “Arthur-Selberg trace formula,” an equation that shows how geometric information can be used to compute arithmetic information.
However, Langlands encountered a significant obstacle when using this formula, as complex sums kept appearing. According to Langlands, these sums were equal, but he could not prove it. He considered it a simple problem and referred to it as a “lemma” – a subsidiary result used to prove more important results. However, no one could prove it, and it became known by a more significant name: “Fundamental Lemma” (FL).
Many leading mathematicians have dedicated efforts to proving FL but have only succeeded in specific cases. Professor Ngo Bao Chau succeeded in proving this lemma in the general case, clarifying long-standing doubts and creating new optimism for mathematical research and many other scientific fields.
Let’s borrow Professor Chau’s words from an interview with Thanh Nien newspaper, where he spoke about FL and the Langlands program:
“The Langlands hypotheses have been a driving force for the development of theoretical mathematics over the past forty years. Many problems that seemed like individual bricks are now reorganized by the Langlands hypotheses into a magnificent architectural structure. Personally, I rank the Langlands hypotheses alongside Euclidean geometry or the invention of Galois groups in solving algebraic equations…
FL is a “lemma” because it is essentially a technical problem. But it is also not merely a lemma because Langlands only proved it in a special case, while the general case was posed as a hypothesis. The term “fundamental” is used because a large part of the aforementioned architectural work would collapse if FL were incorrect. Moreover, proving FL is of great interest to many because its ideas are not confined to the Langlands program but are also intertwined with various issues in theoretical physics.”
We have every reason to expect that Professor Ngo Bao Chau will continue to contribute significantly to the field of mathematics globally and to the nation of Vietnam.
