A Passion: Fourier

[This is an automatic translation of the original post in Spanish and has not been edited yet.]

Long before Sheldon Cooper and his troops in “The Big Bang Theory”, and Ted Mosby in “How I Met your Mother”, there was Ross Geller in “Friends” in that role of the geek friend who has an incomprehensible intellectual passion for the rest. Ross is of course the best of them all, and I have gladly watched the entire series again now that Netflix is ​​streaming it. He did not remember that excellent last episode of the ninth season, in which the whole group travels to Barbados to listen to Ross give a talk at a Paleontology conference. Today when I went back to watch the episode I laughed remembering the time I put my own friends in the same situation.

It was during my last semester of my mathematics degree and the occasion was the support of my undergraduate thesis. For me, that moment marked the end of a job that had lasted a whole year and, for some reason, it seemed most noble to ask family and friends to accompany me for that presentation. I don’t know the exact moment when they will have realized the tremendous mistake they made in accepting my invitation, but surely it was at minute one, when I showed the first slide with the pompous and cryptic title: Construction of biortogonal wave systems of first and second generation and its application in the numerical solution of equations in partial derivatives. For the next hour, I bombarded the audience with a host of equations and graphs, and it’s testament to how much they all love me that no one fell asleep halfway through my chatter.

The “waves” that appear in the title of my undergraduate thesis are closely related to the topic that summarizes one of my greatest passions, and that has connected in some way or another with almost all the topics I have worked on in the last fifteen years: The Fourier Analysis. This is a theory that deals with the possibility, methods and consequences of breaking down a block of information into characteristic frequencies. These frequencies can then be used to understand, manipulate and reconstruct the original information.

I have spent years studying, applying and teaching Fourier, and I know that I could happily spend the rest of my life working in this area of ​​mathematics and still would have only scratched the surface of everything behind it. I have used Fourier to locate earthquake epicenters, obtain images of tissues inside the human body with CT scans, multiply numbers with millions of digits, speed up the simulation of the transitional regime of integrated circuits, and calculate the price of financial derivatives. Fourier is connected with Chebfun, that wonderful project I worked on for four years when I was at Oxford, and Fourier is connected with Artificial Intelligence, the new area in which I am going to focus my career from now on.

Fourier’s analysis is behind the digital music we listen to every day, and in the everlasting movement of the stars. Fourier is studied in Engineering because it is an incredibly useful technique for what electrical, civil and mechanical do. But at the same time, Fourier can detach himself from any mortal bondage to become abstract harmonic analysis and be the focus of study of the purest mathematicians.

Jean-Baptiste Joseph Fourier was born this same week 250 years ago and transformed the world. His theory is fascinating and I can say without blushing that I am passionate about it. I will be glad to invite you all the next time I go to speak in public about it. I know they will not sleep.