A 17 -year -old woman refutes a proposed mathematical conjecture 40 years in the past | Coffee and theorems | Science | EUROtoday

Hannah Cairo had caught a mathematical drawback. In these weeks I did not take into consideration anything and determined to attempt a brand new method. “After months trying to demonstrate the result, I managed to understand why it was so difficult. I realized that, if I used that information in the right way, maybe I could refute the statement. Finally, after several failed attempts, I found the way to build a contract [un caso que no verifica la propiedad estudiada y que demuestra que no es cierta, de forma general]” [el profesor de la asignatura en la que se había planteado el problema] that my proposal was really correct, ”says Cairo.

It turned out to be and, with it, Cairo resolved the so-called conjecture of Mizohata-Takeuchi, a problem proposed in the 1980s, on which the community of the harmonic analysis had been working for decades. Although it was expected to be true – so, other importance results in the field were automatically resolved – the community received the result with enthusiasm. And with surprise: its author was a 17 -year -old girl who had not finished the institute yet.

“When I moved to the US from Nassau [Bahamas, donde nació]I joined the educational system as an institute student, although I took classes at the University of Berkeley. He wrote to the teachers, told them what books he had read on the subject and asked them if he could attend his classes. Many told me yes, among them Zhang, “he says.” One day, he proposed to try a special case of the conjecture, much simpler, as a homework. As an optional part, he raised the original conjecture. And I became obsessed with her, ”he provides.

The conjecture of Mizohata-Takeuchi is included in the field of harmonic analysis, which tries to break down functions in simpler components, such as sinusoidal functions. Today, it is a very hot research area and has also become a fundamental tool in numerous applications, from the compression of digital audio and video files to the design of telecommunication systems.

The harmonic analysis was born in the early nineteenth century with the work of the French mathematician Joseph Fourier on the study of heat function, an equation in partial derivatives that describes the diffusion of heat in a solid body. His revolutionary idea was to decompose this complex function as a sum of breasts and cosenos. The technique, known as Fourier series, opened the door to a new way of understanding physical and mathematical phenomena. “In the idea of harmonic evaluation, all the pieces consists of waves. Anything with them could be constructed, if an acceptable variety of the suitable waves is used,” describes Cairo.

In Fourier’s restricted analysis, which type of objects can be obtained, if only a small set of waves are split. “You can solely construct sure issues, and it is vitally obscure which. The conjecture of Mizohata-Takeuchi states that, if solely sure kinds of waves are used, a kind that’s fabricated from traces is obtained,” he explains.

“Once the primary contract was obtained, I attempted to reformulate your entire drawback within the frequency area. And I noticed how my building was seen. Then I noticed that, in actuality, there was one other a lot simpler approach to design a counterexample,” he declared, satisfied, in one of the rooms of the San José residence, in El Escorial, where from June 9 to 13 International Congress on Harmonic Analysis and Partial Derivative Equationsorganized by the Institute of Mathematical Sciences (ICMM) and the Autonomous University of Madrid. The event, known as El Escorial meetingshas become, in its almost 50 years of history, one of the most prestigious in the area.

This is Cairo’s first international scientific trip. Two weeks ago he landed in Barcelona and, since then, this is his fourth congress. “It is a superb expertise to spend with different individuals who like arithmetic,” he says. In Congress in El Escorial he taught one of the program’s talks. And, far from feeling self -conscious, he enjoyed it. Cairo likes to speak in public. He loves to teach other students – occasions, greater than her. His vocation, he says, is “to assist different individuals, make them blissful.” And, since he has memory, mathematics fascinates him.

He began to read complicated books of the discipline on his own. “I always wanted to be mathematics, but I did not really know what it meant until I learned abstract algebra, through books. It is curious, because the abstract algebra is on the opposite side of the mathematics that I do now. Actually, at first I thought I would dedicate myself to the theory of numbers. With thirteen or fourteen years I wrote an article about the theory of numbers, but I Nobody, ”he recalls, with a laugh.

During the Covid pandemic, the summer camp of the Mathematics Circle of Berkeley – a meeting in which pre -university students try to solve, in a collaborative way, difficult problems of mathematics, similar to the small Institute of Mathematics (PIM) of the ICMAT – had to develop in online format. This allowed Cairo, from Bahamas, to enroll in the course. “Mathematical circles encompass exploring and sharing your concepts with mates; they don’t have anything to do with faculty arithmetic, during which you must memorize. The work is much like portray an image along with your concepts. It isn’t about reaching a tangible goal, however just by understanding issues, asking questions, and it’s also a good way to make mates,” he describes.

The director of the program realized the extraordinary mathematical talent of Cairo – another of the objectives of this type of activities is to identify people with special ability to mathematics and promote their interest and ability – and proposed to be a teacher in the following editions. He did so. Now, at his new university, that of Maryland, where his doctorate begins next, hopes to start his own group.

There, he will continue working, directed by Zhang. “It has helped me so much and I’m very grateful. Beyond his class, which I beloved, he devoted many hours of tutoring,” he recalls. In Spain, the new Mathematics Intensive Program (MIP) of the ICMMA also seeks to detect and accompany trajectories of this type.

Ágata Timón García-Lonongoria She is coordinator of the ICMAT Mathematical Culture Unit.

Coffee and theorems It is a section dedicated to mathematics and the environment in which they are created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which the researchers and members of the center describe the latest advances of this discipline, share meeting points between mathematics and other social and cultural expressions and remember those who marked their development and knew how to transform coffee into theorems. The name evokes the definition of Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms espresso into theorems.”