A mathematician from Peking University has resolved a 50-year-old conjecture. A butterfly's wings have stumped a Fields Medalist.

After 50 years, the major conjecture of the "Ten-Martini Problem" has finally been proven! Lingrui Ge and other scholars from Peking University have found the dual equation in the global theory, unlocking the mystery of a butterfly's wings from a mathematical perspective.

After half a century, the "Ten-Martini Conjecture" has finally come to an end.

This puzzle that connects quantum physics and mathematics aims to solve the mystery of "a butterfly's wings" - the "Hofstadter butterfly".

Famous American mathematician Mark Kac once joked that he would offer ten martinis to anyone who could solve this problem.

Now, Lingrui Ge from Peking University, based on the "global theory" of Fields Medalist Artur Avila, has found a brand - new perspective to break the deadlock.

He joined hands with Jiangong You and Qi Zhou from Nankai University to construct a single, powerful and elegant proof, successfully uncovering the secret of the "dual equation".

This proof was published in the arXiv journal in 2023.

Paper link: https://arxiv.org/pdf/2306.16387

It not only solves the Hofstadter butterfly problem but also further proves the powerful force of abstract number theory in physical reality.

Moreover, based on the improved "global theory", they have successively solved two other key problems.

In this regard, Lingrui Ge said that the mystery behind the "global theory" is like a lighthouse in the dark ocean, guiding us in the right direction.

Where does this problem worth "ten martinis" come from?

An accidental attempt gives birth to a butterfly

In 1974, Douglas Hofstadter, a physics doctoral student at the University of Oregon at that time, went to Regensburg, Germany, with his supervisor to practice German.

At that time, they joined a team of top theoretical physicists, trying to solve a quantum mechanics problem:

How to determine the energy levels of electrons in a crystal lattice placed in a magnetic field?

Douglas won the "Pulitzer Prize" for his book Gödel, Escher, Bach: An Eternal Golden Braid

Douglas was like an "outsider", completely unable to keep up with others' train of thought.

Looking back now, it turned out to be a blessing. He said, "The team was busy proving various theorems, but they had nothing to do with the essence of the problem."

So, he chose to find a different way and tried a more "down - to - earth" method.

Instead of proving theorems, he used an 18 - kilogram Hewlett - Packard 9820A desktop calculator to solve the Schrödinger equation - the core equation of quantum mechanics.

It can explain the behavior of electrons in a specific environment, especially how much energy they have.

In this problem, the Schrödinger equation contains a "variable alpha" - the product of the magnetic field strength and the area of the lattice unit, which summarizes the information of the force acting on the electrons.

When alpha is a rational number, either an integer or a fraction, it is difficult but feasible to solve;

but once alpha is an irrational number, that is, it cannot be expressed as a fraction, the problem becomes tricky.

Douglas didn't fight against irrational numbers like others. Instead, he started with rational numbers, programmed to calculate them one by one, and output results day and night.

The vertical axis represents the magnetic flux, and the horizontal axis represents the electron energy levels

Finally, he pasted these "energy level values" on graph paper and outlined an amazing pattern with a pen.

Because it looks like a butterfly's wings, it is known as the "Hofstadter butterfly".

This pattern reveals a fractal structure: as the denominator of the rational number alpha increases, the forbidden bands between the energy level bands increase.

What's even more amazing is that the tiny local parts of the pattern are surprisingly similar to the overall shape.

The fractal pattern formed by electron energy levels

He keenly noticed that there must be a profound mathematical truth behind this fractal - the Cantor set (Georg Cantor).

Douglas noticed that as the rational number alpha value gets closer and closer to an irrational number, the set of allowed energy levels - that is, the ink bands on each row of the butterfly diagram - also becomes more and more like a Cantor set.

Therefore, he boldly hypothesized that when alpha is an irrational number, the electron energy levels may form a true Cantor set.

The core idea of the Cantor set: Take a line segment, divide it into three equal parts, and then erase the middle part to get two line segments separated by a gap. If this process is carried out infinitely, an infinite set of points will be obtained, scattered on the number line like dust

However, Douglas's discovery was not immediately recognized.

His colleagues laughed at his method as "turning straw into gold", and even his supervisor denounced it as "numerology" and threatened to cut off his research funding.

But Douglas remained unmoved. His intuition told him that this "butterfly" was extraordinary.

The "bet" of ten martinis

Years later, two famous mathematicians reached the same conclusion from a completely different perspective.

Barry Simon and Mark Kac were studying the mathematical object of "almost - periodic functions" at that time.

Different from periodic functions, their collections are infinitely close to repeating but never repeat.

In 1981, Barry and Mark had lunch together and discussed the "Schrödinger equation" that Douglas was trying to solve.

They found that when alpha is an irrational number, the equation exactly becomes an "almost - periodic function".

Based on their understanding of "almost - periodic functions", they concluded that the electron energy levels may indeed form a Cantor set, further confirming Douglas's conjecture.

However, proving this conjecture was extremely difficult. At that time, Mark Kac declared that he would treat anyone who could prove it to ten martinis.

This challenge was thus named the "Ten - Martini Conjecture" and became a major unsolved problem in the mathematical community.

Over the years, mathematicians have made continuous progress and gradually proved that the conjecture holds for some (but not all) irrational numbers alpha.

In 1982, Barry announced a such phased result, and Mark fulfilled his promise with three martinis. Unfortunately, when Mark passed away in 1984, the problem was still not completely solved.

A complete proof worth ten martinis did not come until 20 years later.

After 20 years, a Fields Medalist proves it

In 2003, Svetlana Jitomirskaya had just given up the goal of making the "Ten - Martini Conjecture" her lifelong career.

For years, she had been focusing on studying the "almost - periodic functions" in the Schrödinger equation, but a year ago, her competitor Joaquim Puig got ahead of her.

Joaquim proposed an elegant argument based on the techniques she had published earlier, proving all cases except for a few types of irrational numbers alpha.

Svetlana recalled that all the most difficult work was in my proof, but he got the credit.

Just when she was disheartened, 24 - year - old Artur Avila proposed to cooperate to solve the remaining alpha values.

The two worked together and finally published the proof in 2005, which was published in the Annals of Mathematics.

Paper link: https://arxiv.org/pdf/math/0503363

Therefore, Artur also won the Fields Medal. They decided to fulfill the "ten - martini" agreement themselves and celebrated with a good drink.

However, this proof is actually not perfect.

It only applies to specific irrational numbers alpha and needs to be combined with a phased proof by predecessors to claim that the problem is solved.

It's like a "patched - up dress", lacking integrity and elegance.

More importantly, this proof is based on simplified assumptions about the electron environment, far from the complexity of the real world.

Simon Becker, a mathematician at ETH Zurich, questioned, "You've just verified an ideal model, but what does it have to do with the real world?"

In the real world, the atomic arrangement patterns are more complex, and the magnetic field is not completely constant.

Once the Schrödinger equation is adjusted, the "Ten - Martini" proof fails, seemingly implying that those beautiful fractal patterns - the "Hofstadter butterfly" - are just mathematical coincidences.

Even the proposer Douglas himself wrote in his book Gödel, Escher, Bach, "If the butterfly is really observed in the experiment, I'll be the most surprised person in the world."

By 2013, a group of physicists at Columbia University measured the electron energy levels of two layers of ultra - thin graphene in a magnetic field and successfully captured the "Hofstadter butterfly".