From Herrera to Riemann passing by Perelman (v2)

Nota: Al parecer el pequeño texto que escribí hace un tiempo sobre Rutilio Herrera y la Hipotesis de Riemann ha tenido buenos resultados. Por sugerencia del mismo Rutilio y varios amigos, Javier Alfaro se hizo a la tarea de traducir el texto.

Many must have followed the news about the curious story of Genius Mathematician Grigori Perelman and the fame (or infamy) he achieved by giving solution to the Poincaré conjecture. After solving one of the greatest mathematical problems in history, the media announced with flying colors, Grigori’s refusal to accept any recognition, including the two most prestigious awards of the Mathematical world: The Fields Medal and the 1 million dollar prize from the Clay Mathematics Institute. To make the story even more striking, just after writing the three articles on which he resolved such problem this numerical genius decided to turn his back on the Mathematical world and now spends his days playing ping-pong in an apartment at his native Saint Petersburg.

Something that comes to my mind every time I stump by any news on Perelman and his mathematical achievement is of one my high school mathematics professors: Rutilio Herrera. For those skilled on the pen, the story of his life certainly has the caliber to be documented much more thoroughly than the short summary I give below:

It was in 1992 while on my 3rd year of high school, when Rutilio came to our school as a new Mathematics teacher. At the time he taught at the School of Mathematics of the University of Costa Rica and in our school. I remember well his first lesson when all the students looked at him surprised while he filled the board with mathematical notation, symbols unknown to us at that time. By the second week of lessons we had to ask him to please write in Spanish or to at least give us an introductory class to those strange symbols.

Rutilio had a charisma hard to find in high school mathematics teachers. His lessons were interspersed with stories about the difficulties he encountered while being a mathematics university student in his native El Salvador where, as the country was in the middle of a civil war, he and his classmates were mocked by both the guerrilla and the government soldiers. In his lessons he often spoke with great enthusiasm about the marvels of equations, Calculus, about the many real world applications of numbers and from time to time he also commented about some Riemann hypothesis to which according to him, he spent several years of his life researching for its solution.

We were only in the 3rd year and very little we understood about such Riemann hypothesis and even less about the efforts that Rutilio had to go trough to resolve it. I remember that at that time he told us that he had had the opportunity of presenting his solution in two mathematical conferences, one in Italy and the other in Japan. Back then Wikipedia didn’t exist and we didn’t have the slightest idea about the significance of that hypothesis nor the one million dollars prize that the Clay Mathematics Institute awarded to whom is able to verify it. With the advent of the internet and Calculus lessons at the University I realized the importance of Riemann and his hypothesis and also that Rutilio’s efforts have not been documented nor discussed.

Along with some of my high school generation classmates I have remained in contact with Rutilio. We know about his resignation to his teaching positions both at our former school and at the university and that now he devotes his time to his Academy of Mathematics Euclides in Moravia – San Jose – Costa Rica. Every time we meet we ask him about the recognition of his work on Riemann but his answer is always the same: “Due to the innovation of the approach I used to address the hypothesis, mixing two very different areas of mathematics (Singular integral equations with a Cauchy kernel and Analytic number theory) still no mathematical Journal wants to publish the manuscript.

For an article to be published in a journal the manuscript has to go through the process of editing and approval of a group of scientists who are authorities in the areas involved in the work (what is also called peer-review). Depending on the article and the issues involved this editing and approval process can last from few months to several years. For example in case of Rutilio’s work the answer from the editor of the journal The Mathematical Intelligencer took three years to arrive. The problem here is that the more innovative the focus of the work is, the harder it will be for the group of reviewers from the journal to accept it. In Rutilio’s case - as he has told us - his approach using Singular integral equations with a Cauchy kernel (applied mostly to problems in physics) comes to make a huge difference to the Analytic number theory which is the most common approach used in the previous attempts to verify Riemann’s hypothesis.

A few years ago, while I was reading the first news about Grigori Perelman and his solution to the Poincaré conjecture, I noticed that Perelman rather than publishing his article in a journal he used arXiv, a very popular online database where many mathematicians publish their manuscripts without being peer-reviewed. Thinking about that, I wrote Rutilio commenting on the possibility of sharing his work using arXiv. The problem came when we realized that an invitation is required to upload a manuscript to arXiv, and the invitation has to come from another mathematician with access to the database. So far neither Rutilio, me nor my other friends in Costa Rica that also wish to see his work fulfilled have managed to obtain that invitation.

As I said earlier this is merely a summary of the memories I have of conversations with Rutilio. I know that he has tried through Franklin Chang to obtain some kind of reaction from the mathematicians involved with Chang’s NASA projects. I have also tried to put Rutilio in contact with friends of friends, phd students in mathematics at UC Berkley but efforts have not been entirely successful.

Finally, I want to clarify that I have no idea of the validity of the approach that Rutilio used to solve such an important mathematical problem - my mathematical knowledge is pretty basic - although I share Rutilio’s view, that as his work represents a novel approach, it is important for the international mathematical community to study and discuss it.

If you are interested in the story, you can write Rutilio Herrera.


  1. Me acaban de avisar que Rutilio passed away.. Hasta luego querido amigo siempre voy a recordar cuando me entregaste una copia de tu trabajo sobre la Hipótesis de Riemann que curiosamente tienen tus siglas al revés.


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