- Margarita Rodriguez
- From BBC News World

**A species of butterfly has captured the attention of math teacher Svetlana Zhitomirskaya for years. This admiration was one of the factors that prompted her to delve deeper into a mathematical problem presented in 1981.**

“The Hofstadter butterfly is a very interesting thing to observe,” she says.

It is a graphic representation of a fractal group, which was created in the 1970s by the American scientist Douglas Hofstadter, who played an important role in quantum mechanics.

But there was a specific point I made to this issue: the “meaningful ideas” I developed in this area. She also highlights that Guess “has a very catchy name.”

Zhitomirskaya helped solve the “problem of 10 martini”. This name came after the Polish mathematician Mark Kac presented his 10 martini to his solver.

Kak could not estimate the professor’s achievement because he died in 1984. But fellow American Barry Simon gave the conjecture the name he popularized.

I asked the researcher who lives in the United States: “Have you ever had a martini?” “I had a martini,” she replied with a laugh, “but not because of that problem.”

This is the story of one of the most important mathematicians of our time. His contributions to mathematical physics and dynamical systems are widely recognized.

In July 2022, Zhitomirskaya was awarded the first Olga Alexandrovna Ladyzhinskaya Prize, in a joint session of two conferences held during the International Congress of Mathematics.

## Daughter of mathematicians

Svetlana Zhitomirskaya was born in Kharkiv, Ukraine (then part of the Soviet Union) in 1966.

She speaks fondly of her mother, the pioneering mathematician Valentina Borok (1931-2004), who worked on partial differential equations and, in 1970, became Ukraine’s only professor of mathematics.

“She was so bright that I knew I wasn’t as bright as she was,” she says. “Somehow, I didn’t think I could succeed in mathematics because it was too difficult for women at the time.”

She says, “Of course, especially as a woman, I had to stand out a lot, not because of discrimination, but because despite all the communist propaganda that women were equal, society as a whole was very traditional and expected for women to take care of the family and home environment.”

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

Although this inspired her, she knew that her mother did not want her to go down the path of mathematics. Her father, also a mathematician, did not contradict her.

“My parents used to work together,” she says. “When I was little, it was as if they somehow tried to dissuade me from studying math because they thought it was too hard for a girl.”

“A while ago I asked my father why they were discouraging me and not my brother. He replied, ‘It was your mother’s idea. ‘ I think that’s right, it was her idea to try to steer me toward other things.” , Says.

## Some kind of miracle

Zhitomirskaya loved literature and philology. For her, however, the Soviet Union was not the ideal place to pursue this passion, as these studies were steeped in communist ideology.

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

“It was a great environment for a student who was ready to take in everything and willing to study hard,” she says. “And the entry was kind of a miracle because basically they didn’t accept Jews.”

Zhitomirskaya says she prepared well for the admissions process because Jewish applicants were treated very differently.

“They posed very difficult problems, practically impossible to solve,” she says. “So I spent my last year of high school preparing for this exam.” However, I thought it wouldn’t pass.

“Somehow they didn’t realize I was Jewish,” she says. In the documents, it is listed as Ukrainian. Zhitomirskaya was accepted and, at the age of 16, made use of all available educational resources: “fantastic conferences and seminars”.

“I really fell in love with math and never looked back,” she recalls. “I remember, in my sophomore year of college, I thought I couldn’t imagine studying anything other than math.”

Years later, an academic opportunity presented itself to her husband, a physical chemist, and the couple moved to the United States. There, she took a temporary position as a part-time professor at UC Irvine and pursued her research.

Zhitomirskaya is currently studying at that institution, and was recently appointed Professor at the Georgia State Institute of Technology in the United States.

## Mathematical Physics

One of the focal points of her field of study, Zhitomirskaya explains, is substantiating the guesses made by physicists, “ideas that have long been understood.” But it also does the opposite: “Sometimes we disprove them and prove them wrong, and sometimes we make new predictions related to physical models.”

“It’s very exciting because sometimes real-life connections are made, but not always,” she says. “Concretely, I work in the field of quasi-periodic operators.”

These operators are related to quantum mechanics and the “10 Martini Problem” is part of this fascinating field.

Since the 1990s, Zhitomirskaya has worked on many aspects of this conjecture. I was able to find several pieces of the puzzle and have published their findings. Until 2003, Spanish mathematician Joaquim Puig “made substantial progress on this problem.” He mentioned Zhitomirskaya’s work in his research.

“He noticed something very beautiful,” she says. “It felt like a small addition to my previous work, but it was a great observation and I was a little disappointed in myself that I didn’t see that way about the problem.”

## “all parameters”

Less than a year later, a “very young” Brazilian mathematician contacted her. This boy, in 2014, would go on to win the Fields Medal, also known as the Nobel Prize in Mathematics.

“Artur Avila wrote to me because he wanted to visit me to work on this problem,” she says. “I’ve already seen his name because he has published two excellent articles.”

Jitomirskaya recalls saying, “The problem is not completely solved until you unscrew all the parameters.” Avila had indicated in one of her posts that she had hinted that she might get another result for the “remaining criteria”.

“And he told me that if I could really do it, we could totally work it out,” she says. “He said it could be done, but it would be very difficult, technical and time consuming.” But Avila persuaded her.

When they began working on “this very difficult art show”, they realized they would have to “invent other ways”. And in the process, they have developed innovative tools, techniques and methods that are admired by experts.

They proved the conjecture and published the result in the prestigious 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 Supreme Council for Scientific Research (CSIC) in Spain. He is familiar with the work of Jitomirskaya, whom he met at many conferences.

“It’s always good to talk to her and listen to her presentations,” Peralta told BBC News Mundo (the BBC’s Spanish-language service).

He recalls a conference in China where mathematics showed a Hofstadter butterfly, which represented the spectrum of the operators she was studying. Peralta shows that these operators appear in certain models that attempt to describe physical phenomena of the quantum type.

He explains, “Schrödinger operators appear in many contexts of quantum mechanics, and Zhitomirskaya mainly studied those that arise in the context of the motion of electrons exposed to magnetic fields perpendicular to the dynamics of electrons.”

They are known as quasi-periodic Mathieu operators.

“In general, a quantum mechanics operator is a mathematical object, a mathematical rule, that assumes a function with different values and returns another characteristic function,” Peralta explains.

The key is to understand the spectrum from a physical point of view, i.e. to see which functions an operator, when applied to, returns the same function. And this, according to the researcher, is one of the big differences (among many others) between classical and quantum physics.

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

“But in quantum mechanics, there are many things that are quantified, they can’t assume any value and can only assume a series of discrete values,” Peralta explains. “This phenomenon and Heisenberg’s uncertainty principle (i.e., the fact that certain magnitudes cannot be precisely measured) point to the main difference with respect to classical physics.”

## Width

In the 1960s, physicists noticed that the values that this type of operator can assume depend on frequency, that is, on the change in the spectrum when the parameters differ.

They noticed it when hesitating [um número] Irrational, the spectrum had a very strange fractal structure, what is known as the Cantor group. And that’s what’s presented mathematically in the ten martini statement,” says Daniel Peralta.

The problem is proving something physicists have already noticed – that when the frequency, i.e. the magnetic field strength of this type of operator is an irrational number, the spectrum is a Cantor group.

Many researchers have been working on this problem since the 1980s and 1990s, and Puig has made great progress, but “the culmination of all this work, for years and many people, is the exposure Avila and Zhitomirskaya got.”

“They confirmed the original conjecture: for all irrational frequencies, the spectrum of quasi-periodic Mathieu operators is the Cantor group,” Peralta says. Thus the “problem of 10 martini” was finally solved.

** **This report is part of the special *BBC 100 Women*which each year highlights 100 inspiring and influential women around the world.*

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