As a child, Marguerite Hoffman was frightened by the thought of the infinity of the universe, until, while browsing through a book in the library, she came across a graphic representation of the famous Goldbach conjecture, whose statement was formulated by Christian Goldbach in 1742. Thus, she found a way to put order in infinity, to which she would dedicate her life. Marguerite is the protagonist of the film Marguerite’s theorem -released last week in Spain-, a 25-year-old mathematician who is in her final year of doctoral thesis at the École Normale Supérieure (ENS) in Paris (France).
Written and directed by Franco-Swedish director Anne Novion and starring Franco-Swiss actress Ella Rumpf, winner of the César Award for Best New Actress for this role, the film depicts the profession of mathematics research, at the highest level in academia, with the fear of failure ever present. The precision of the mathematical details that appear throughout the narrative is made possible by the scientific advice of Ariane Mézard, professor at the Sorbonne University (France).
The story begins by focusing on Marguerite’s great talent and dedication to mathematics: “A mathematician of your level is rare,” says her thesis advisor, Laurent Werner. This expression can have a double interpretation: on the one hand, there are few people with an extraordinary talent for mathematics, as is the case in other fields; on the other, there is a clear shortage of women in this field. Novion chose Marguerite as the protagonist to remind us that there are also women doing mathematics, who have been preceded by others, who have obtained important results, such as Sofia Kovalevskaya, Emmy Noether, Karen Uhlenbeck, Maryan Mirzakhani, Maryna Viazovska.
However, talent and effort are not incompatible with mistakes. Marguerite sees how her work of the last three years falls apart when she presents it to her colleagues, when one of them detects an error. Although, at first, the young researcher does not take this defeat well, this happens frequently in mathematics. To prove any mathematical result, it is necessary to be able to check all the steps. Very sophisticated techniques and ideas are used in reasoning that can only be understood by a few specialists in the subject, who are the ones who will validate the proof. An error or a step not explained correctly can invalidate the entire construction and the immense edifice of the demonstration collapse. When a proof is verified step by step, it gives rise to a theorem or proposition.
Marguerite is writing her thesis in the area of analytical number theory. With it, she seeks to prove Szemerédi’s theorem, related to arithmetic progressions in sets of integers, from a new perspective. Achieving this result could bring her closer to her final goal: the demonstration of Goldbach’s conjecture. This is one of the most famous open problems in the discipline, whose statement is quite simple: every even number greater than 2 can be written as the sum of two prime numbers, for example, 6=3+3, 8=5+3 or 112=31+83. Working on such complex problems is, therefore, a great challenge that can lead to making mistakes, something that great mathematicians have not been spared from.
This turning point in Marguerite’s career shows the importance of teamwork, and that constant review of research is essential to progress. In fact, before presenting her work, she suspects that there is a flaw in one of the parts of her thesis, but she is unable to detect it, so she asks her thesis supervisor for help, which he refuses. This collaborative aspect is reinforced when Marguerite later asks for help from Lucas Savelli, the same person who had found her flaw: “With Goldbach I couldn’t do it alone,” he tells him. Marguerite and Lucas set out to solve the problem, almost obsessively and outside of academia.
The story also shows us how mathematics is a constant in our daily lives. In the game of mahjong – Chinese poker – Marguerite finds the inspiration to advance in Szemerédi’s problem. She manages to find a relationship between the two, which allows her to prove the existence of certain arithmetic progressions in prime numbers. Although this is a cinematic device, something similar can happen in the path of research, that there is a kind of serendipity and unimaginable forks are found that may lead to the final point, in the most unexpected place.
David Martin de Diego He is a scientific researcher at the CSIC at the Institute of Mathematical Sciences (ICMAT) and scientific director of the Mathematical Culture Unit at ICMAT.
Laura Moreno-Iraola is a journalist and member of the ICMAT Mathematical Culture Unit.
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