- Margarita Rodriguez
- BBC News World

**There is one type of butterfly that has fascinated Professor Svetlana Zhitomirskaya for years.**

This led her to look at a mathematical problem presented in 1981.

“The Hofstadter’s butterfly is a very pleasant thing to look at,” he says.

It is a graphical representation developed by the scientist Douglas Hofstadter in the 1970s of a fractal set and played an important role in quantum mechanics.

But one decisive aspect drew her to the problem: “some remarkable ideas” she had developed in the field.

Plus, he says, Assumption “has a very impressive name.”

Mathematician helped solve the “Ten Martini Problem,” a name coined after mathematician Mark Cocke offered ten martinis to anyone who could solve it.

Cock doesn’t know how to appreciate Zhydomirskaya’s exploits. He died in 1984, but it was his colleague Barry Simon who popularized the conjecture by giving his name to it.

“Have you ever had a martini?” I ask a US-based researcher.

“I’ve had martinis before, but not for this problem,” she laughs.

This is the story of one of today’s greatest mathematicians, whose contributions to mathematical physics and dynamical systems are widely recognized.

Most recently, in July, Olga Alexandrovna Ladyzhenskaya received the first prize at a joint session of two satellite conferences of the International Congress of Mathematicians.

## Among mathematicians

Zhitomirskaya was born in 1966 in Kharkiv, Ukraine.

He was raised by his mother, Valentina Borok, a leading mathematician who worked on partial differential equations and became the only professor of mathematics in Ukraine in 1970.

“She’s very bright, and I know I’m not,” he says.

“In a way, I didn’t think I could do well in math because it was so hard for girls.

“It was clear that you, especially as a woman, had to be very exceptional, but not because there was discrimination. Despite communist propaganda that women were equal, all society was very traditional and women were expected to take care of the family and the domestic sphere.”

“My mother always told me that family is the most important thing.”

And, even though she was an inspiration, she knew her mother didn’t want her to go down the math path.

Her father, a mathematician, did not contradict her: “My parents worked as a unit”.

“When I was little, they tried to talk me into becoming a mathematician because they thought it was too difficult for a girl.”

“Recently I asked my dad why he was encouraging me and not my brother and he said, ‘I think it was your mum’s idea’ and I think that was true, she had the impression that they were trying to steer me towards other things.

## Like a “miracle”.

Zhitomirskaya loved literature and linguistics, but for her, the Soviet Union was not the best place to pursue these interests, as these fields were strongly steeped in communist ideology.

Then she fell in love with mathematics when she began to study it in depth, “which only happened when I entered Moscow State University.”

“It was an incredible environment for a student who was willing to absorb everything and study hard.

“It was a miracle that I was accepted because they didn’t accept Jews.

He was well prepared for the admissions process, as Jewish applicants were treated very differently.

“They were given very difficult problems that were basically impossible to solve. So I spent my last year of high school preparing for this test.”

Still, she thought she wouldn’t pass.

“Somehow they didn’t realize I was Jewish. She says she was listed on the document as Ukrainian.

He was admitted, and the 16-year-old took advantage of all the educational resources, including “wonderful lectures, seminars.”

“I really fell in love with math and never looked back. In second grade I thought I couldn’t imagine studying anything but math.”

Years later, after her husband, a physical chemist, got an educational opportunity, Zhitomirskaya accompanied him to the United States.

Irvine finds a temporary job as a part-time lecturer at the University of California and continues his research.

He is currently a professor at the institute and was recently appointed as a professor at the Georgia Institute of Technology.

He explains that his field’s strength is testing physicists’ assumptions, which are “long-understood concepts.”

But it also does the opposite, “sometimes we reject and prove them wrong, sometimes by making new predictions about physical models.”

“It’s really exciting because sometimes you relate to real life, but not always.

“Specifically, I work in the field of part-time operators.”

It is about quantum mechanics and the “10 martinis problem” is part of this fascinating field.

Since the 1990s, Zhitomirskaya has worked on various aspects of this hypothesis, obtaining several “pieces of the puzzle” and publishing her results.

In 2003, the Spanish mathematician Joaquim Puig made “an important advance on this problem”. Indeed, in his research, he cited the works of Zhitomirskaya.

“He made a very good observation. It seemed like a small addition to my previous work, but it was a very clever observation and I felt a little bad about myself for not seeing that path into the problem.”

## “All Settings”

A year later, a “very young” Brazilian mathematician contacted her (years later, in 2014, this boy would win the Fields Medal (also known as the Nobel Prize in Mathematics).

“Arthur Avila wrote me that he wanted to meet me to work on this problem. I had seen his name before because he had published some excellent articles.

“The problem is not completely solved until you determine all the parameters,” Shydomirskaya recalled her telling him.

Avila mentioned to her that he saw one of her posts that he could get another result for the “remaining parameters.”

“He told me that if I could do that, I could completely solve the problem, and I told him that it was possible, but it was very difficult, technical, and it would take a long time.”

But Avila insisted.

When they started working on a “very difficult technical demonstration,” they realized they had to “find other ways.”

In the process, they have developed new tools, techniques and approaches that are admired by scholars.

They proved the conjecture and published the result in the prestigious journal Annals of Mathematics in 2009.

## What’s the problem?

Daniel Peralta is a researcher specializing in dynamical systems at the Institute of Mathematical Sciences (ICMAT) of the Spanish National Research Council (CSIC).

He is familiar with Zhitomirskaya’s work and has met her at several conferences. “It’s always great talking to him and listening to his presentations,” he told BBC Mundo.

In fact, he recalled a conference in China where the mathematician displayed Hofstadter’s butterfly, which represented the spectrum of operators he was studying.

These operators, Peralta explains, “appear in certain models that seek to describe quantum physical phenomena.”

“Schrödinger operators appear in many quantum mechanical contexts, and Zhitomirskaya mainly studied their appearance in the context of the motion of electrons subjected to magnetic fields perpendicular to the electron’s velocity.”

They are called half-term Mathieu operators.

“In general, an operator in quantum mechanics is a mathematical object, a mathematical rule, which takes a function of different values and returns a different function”.

The main point is that, from a physical point of view, the spectrum, i.e., for any function, when the operator is applied, it gives the same function.

This, the researcher explains, is one of the big differences (among many) between classical physics and quantum physics.

For example, in principle, in classical physics, the speed of an electron, a particle, can take any value.

“Now, in quantum mechanics, there are many measured objects that can take no value at all, only a series of discrete values. This phenomenon, as well as Heisenberg’s uncertainty principle (i.e. some truth. Quantities cannot be precisely measured) constitute the main difference from classical physics. signifies”.

## Demonstration

In the 1960s, physicists realized that the values this type of operator could take depended on frequency, meaning that the spectrum changed as the parameters varied.

“When the frequency was an irrational number, they noticed that the spectrum had a very strange fractal structure called the Cantor set.

“It’s mathematically stated in the Ten Martini Statement.

The problem is – what physicists have already noticed – when the frequency, i.e. the intensity of the magnetic field for this type of operator is a proportional number, the spectrum is a Cantor set.

Since the 1980s and 1990s, many researchers have studied this issue.

Puig has made a breakthrough, but “the culmination of these works, so many years and so many people, is the evidence provided by Avila and Zhitomirskaya.”

“They prove the original conjecture: for all irrational frequencies, the spectrum of half-period Mathieu operators is a Cantor set.

Thus, the “Ten Martini problem” was finally solved”.

**The report is part of a BBC special dedicated to the world’s 100 most inspiring and influential women in 2022.**