- Margaret Rodriguez
- From BBC News World
A kind of butterfly has attracted the attention of mathematics teacher Svetlana Jitomirskaya for years. And this admiration was one of the factors that led her to delve into a mathematical problem presented in 1981.
“The Hofstadter butterfly is an extremely pleasant object to observe,” he says.
It is the graphic representation of a fractal set, created in the 70s by the American scientist Douglas Hofstadter, who played an important role in quantum mechanics.
But there was one defining point that brought her to this problem: “significant ideas” that she had developed in this field. And he also points out that the conjecture “has a very attractive name.”
Jitomirskaya helped solve the “10 martinis problem”. This name came after Polish mathematician Mark Kac offered 10 martinis to whoever solved it.
Kac could not appreciate the success of the professor because he died in 1984. But his American colleague Barry Simon gave the conjecture the name that made it popular.
“Have you ever had a martini?” I asked the researcher, who lives in the United States. “I had a martini, but not because of this problem,” he laughs.
This is the story of one of the most important mathematicians of our time. His contributions to mathematical physics and dynamical systems have been widely recognized.
In July 2022, Jitomirskaya was awarded the first Olga Alexandrovna Ladyzhenskaya Prize, in a joint session of two lectures held during the International Congress of Mathematicians.
daughter of mathematicians
Svetlana Jitomirskaya was born in Kharkiv, Ukraine (then part of the Soviet Union) in 1966.
She speaks with admiration of her mother, the important mathematician Valentina Borok (1931-2004), who worked on partial differential equations and, in 1970, became Ukraine’s only full professor of mathematics.
“She was so brilliant that I knew I wasn’t as brilliant as her,” she says. “In a way, I didn’t think I could succeed in math because it was so difficult for women at the time.”
“Of course, especially as a woman, I had to stand out a lot,” she says, “not because there was any discrimination, but because, despite all the communist propaganda that women were equal, society as a whole was extremely traditional and expected women to women to take care of the family and the domestic environment”.
“My mother always told me that family was the most important thing,” recalls Jitomirskaya.
Though it was her inspiration, she knew her mother didn’t want her to go down the math route. And her father, also a mathematician, did not contradict her.
“My parents worked together,” she says. “When I was little it was like somehow they tried to dissuade me from taking up math because they thought it was too hard for a girl.”
“A little while ago I asked my father why they discouraged me and not my brother. He replied: ‘It was your mother’s idea’. I think it’s true, that it was her idea to try to guide me towards other things “. , she says.
“Some Kind of Miracle”
Jitomirskaya was fond of literature and philology. But for her, the Soviet Union was not the ideal place to pursue this passion, as these studies were so steeped in communist ideology.
She then fell in love with mathematics when she began to study it thoroughly, which only happened when she entered Moscow State University.
“It was an amazing environment for a student who was ready to accept anything and willing to study hard,” she says. “And getting in was kind of a miracle because they basically didn’t accept Jews.”
Jitomirskaya says she prepared very well for the admissions process because Jewish applicants were treated very differently.
“They presented very difficult problems that were practically impossible to solve,” he says. “So I spent my senior year of high school preparing for this exam.” Still, she didn’t think it would pass.
“Somehow they didn’t realize I was Jewish,” he says. In the documents she was listed as Ukrainian. Jitomirskaya was admitted and, at the age of 16, she took advantage of all the educational resources available: “incredible lectures, seminars”.
“I really fell in love with math and never looked back,” she recalls. “I remember, in my sophomore year of college, thinking I couldn’t imagine studying anything but math.”
Years later, an academic opportunity presented itself for her husband, who is a physical chemist, and the couple moved to the United States. There, she took a temporary job as a part-time professor at the University of California at Irvine and continued her research.
Jitomirskaya currently teaches at that institution and was recently appointed a professor at the US Georgia State Institute of Technology.
Jitomirskaya explains that one of the central points of her field of study is to prove the conjectures made by physicists, “ideas that were understood a long time ago”. But she also does the opposite: “sometimes we refute them, we prove them wrong, sometimes we make new predictions related to physical models”.
“It’s very exciting because real-life connections are made sometimes, but not always,” she says. “More 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, Jitomirskaya has been working on different aspects of this conjecture. She has managed to find many pieces of the puzzle and has published the results of her own. Until, in 2003, Spanish mathematician Joaquim Puig “made a fundamental breakthrough in this problem.” And he mentioned Jitomirskaya’s work in finding her.
“He noticed something very beautiful,” according to her. “It seemed like a small addition to my previous work, but it was a brilliant observation and I was a little disappointed in myself that I didn’t see the problem that way.”
Less than a year later, a “very young” Brazilian mathematician contacted her. That boy, in 2014, would win the Fields Medal, also known as the Nobel Prize in mathematics.
“Artur Ávila wrote to me because he wanted to visit me to work on this problem,” she says. “I had already seen his name because he had two excellent articles published.”
Jitomirskaya recalls saying, “the problem isn’t completely solved until you decipher all the parameters.” Ávila had noted in one of his publications that he had hinted that he might get another result for the “remaining parameters.”
“And he told me that if I could actually do that, we could solve the problem completely,” she says. “She said it could be done, but it would be very difficult, technical and time consuming.” But Ávila convinced her.
As they started working on “this very difficult technical demonstration”, they realized they would have to “invent other ways”. And in the process, they have developed innovative tools, techniques and approaches that are admired by experts.
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 Superior Council for Scientific Research (CSIC) in Spain. He knows the work of Jitomirskaya, whom he has met at various congresses.
“It’s always so nice to talk to her and listen to her presentations,” Peralta told BBC News Mundo (the BBC’s Spanish-language service).
He recalls a lecture in China where the math proved Hofstadter’s butterfly, which represents the spectrum of operators he studies. Peralta explains that these operators appear in some models that attempt to describe physical phenomena of a quantum type.
“Schrödinger operators appear in many contexts of quantum mechanics,” he explains, “and Jitomirskaya mainly studied those that arise in the context of the motion of electrons subjected to magnetic fields perpendicular to the electron dynamics.”
They are known as quasi-periodic Mathieu operators.
“In general, the quantum mechanical operator is a mathematical object, a mathematical rule, that takes one function with different values and returns another distinct function,” explains Peralta.
The key is to understand the spectrum from a physical point of view, that is, to see which functions the operator, when applied to them, returns the same function. And this, according to the researcher, is one of the great differences (among many others) between classical physics and quantum physics.
In principle, for example, the speed of an electron or a particle can take on any value in classical physics.
“But, in quantum mechanics, there are many objects that are quantified, cannot take on any value, and can only take on a set of discrete values,” explains Peralta. “This phenomenon and Heisenberg’s uncertainty principle (that is, the fact that certain quantities cannot be measured precisely) mark the main difference with respect to classical physics”.
In the 1960s, physicists observed that the values that this type of operator can assume depend on the frequency, ie on the variation of the spectrum as the parameters vary.
“They observed that when attendance was [um número] irrational, the spectrum had a very strange fractal structure known as the Cantor set. And that is presented mathematically in the 10 martini declaration,” says Daniel Peralta.
The problem consists in demonstrating something that physicists had already observed, namely that, when the frequency, ie the intensity of the magnetic field for this type of operator is an irrational number, the spectrum is a Cantor set.
Many researchers have been working on this problem since the 1980s and 1990s. Puig has made great progress, but “the culmination of all this work, of years and of many people, is the demonstration obtained by Ávila and Jitomirskaya”.
“They confirmed the original conjecture: for all irrational frequencies, the spectrum of quasi-periodic Mathieu operators is a Cantor set,” according to Peralta. So the “10 martinis problem” was finally solved.
* This report is part of the special BBC 100 Womenwhich each year highlights 100 inspiring and influential women from around the world.
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