Died of cancer in 2017, but the ideas of the first Nobel Mathematics have just come to make a new life

Iranian Maryam Mirzakhani transformed the field of hyperbolic geometry. But he died at age 40 before he was able to answer many of the questions that interested her. Two mathematics are now resuming the work that the first woman to win the “Nobel Prize for Mathematics” left.

In the early 2000s, Iranian Maryam MirzakhaniA young graduate student at Harvard University began to draw a mathematical universe that challenged everything that was known about geometric intuition.

Mirzakhani began by focusing on “Hyperbolic surfaces”. On a surface of this type, the parallel lines move away from each other instead of staying at the same distance and, at each point, the surface bends into two opposite directions, such as a saddle.

“A hyperbolic surface is a bit like a puzzle that can be assembled locally, but it can never end in our universe,” he explained to, Alex WrightMathematician at the University of Michigan and former postdoctoral colleague of Mirzakhani.

This is because each piece is curved precisely in the form of a saddle. That is, it is possible to fit some pieces, but never to close the surface completely – at least not in our flat and three -dimensional space. This makes hyperbolic surfaces particularly difficult to study. Even basic questions about them remain open.

In short, hyperbolic surfaces have geometric properties so strange that they are impossible to view.

Mirzakhani revolutionized the hyperbolic universe. When he was still in graduate, he developed innovative techniques that allowed him to begin to catalog these forms before revolutionizing other areas of mathematical research.

Iranian mathematics wanted to understand the aspect of a typical hyperbolic surface; And it began to select hyperbolic surfaces at random and study its properties.

Mirzakhani expected to revisit his map of the hyperbolic kingdom later – to fill details and make new discoveries. But before being able to do so, a Cancro da mother. Mirzakhani died in 2017with only 40 years, when I was already developing the necessary machinery.

Since then, two mathematics have caught the wire of their work and have turned it into an even deeper understanding of the hyperbolic surfaces.

Maryam was really a genius

In an article at Arxiv in February, Nalini Anantharamanof Franc Collelge, and Laura Monkfrom the University of Bristol, were based on Mirzakhani’s investigation to prove a radical statement on typical hyperbolic surfaces.

The new study showed that the surfaces that were thought to be rare (and even impossible), after all, They are common “So much so that if we chose a hyperbolic surface at random, it would be guaranteed that it would have certain critical properties.”

The study, which has not yet been reviewed by peers, also suggests that hyperbolic surfaces are still stranger and less intuitive than imagined.

In 2018, just a year after Mirzakhani’s death, Monk began his postgraduate studies with Anantharaman. His first step was to learn everything he could about Mirzakhani’s work on hyperbolic surfaces.

Monk and Anantharaman needed to show that almost all hyperbolic surfaces have a 1/4 spectral interval.

Using the limited formula of Mirzakhani, Monk and Anantharaman saw a way to prove a relatively large spectral interval. “It looked almost a miracle”He said, how much magazine, anantharaman.

“It’s still a mystery to me that it works so well,” he added.

Suddenly, as the same magazine tells, Anantharaman remembered a mail he had received from Mirzakhani a few years before she died, putting a series of questions about the relationship between the spectral interval and geodetic count, which could improve the formula.

However, Anantharaman had a different idea: resorting to a different area of ​​mathematics, called to be inspired.

In early 2023, the two mathematicians wrote an article that described what they had done so far. The following year, they adapted the methods of the mathematician and expert in this matter Joel Friedman And they made a plan to use them.

And proof was finally completed in January, demonstrating that a surface hyperbolic selected selected randomly has probably the maximum spectral interval.

The result tells the mathematicians more about hyperbolic surfaces than they ever knew – thanks to Maryam Mirzakhani.

“I’m sad that she can’t see the results of this study,” Wright lamented about.