Throughout life we encounter remarkable individuals who shape our future in profound ways. I write these lines as a brief tribute to three teachers who, at different stages of my life, played a decisive role in my development as a mathematician. I include a few anecdotes that I hope will make reading more enjoyable.
My father understood the importance of mathematics, and although both my older brother and I had shown talent in the subject, he arranged private lessons for us when I was eleven. Our teacher, Mr. Juan Valero, was a self‑taught man who had never taught before. His method was simple: after a short explanation —never more than ten minutes— in which he always justified the ideas and processes with impeccable logic, he would open the book and have me work through every problem in the section. He sat at the same round table, attending to several students at once, and whenever I asked a question, he would respond with another question that, once considered, guided me to the solution. His Socratic approach and his habit of breaking problems into smaller ones introduced me to the art of problem‑solving.
In Guadix, the town where we lived, people called him “the madman,” because he had suffered a nervous breakdown while preparing for a notoriously difficult civil‑service exam. At the time, it was common for a thousand candidates to compete for only a few dozen positions. But once teachers at our school began commenting on our unusually strong preparation, students flocked to him, and he eventually made a living from teaching.
I grew up in Francoist Spain, where before 1970 only 28% of students entered secondary school and fewer than 5% reached university. Those of us from small towns had to travel to the provincial capital at the end of each academic year to take one‑hour exams in every subject—exams that determined whether we passed or failed. Very few survived that system. Thanks to a scholarship and my parents’ enormous sacrifice, I completed upper secondary school as a boarding student in Granada. My classmates were exceptional; all of them went on to earn university degrees, and I am certain any one of them would have succeeded in graduate studies in the United States. I am delighted that, thanks to WhatsApp, most of us remain in touch.
The second teacher I must mention is Mr. Manuel Bravo, who taught me mathematics in the Pre‑University course, which at the time followed secondary school and preceded university. The curriculum was a mixture of number theory, combinatorics, spherical geometry, and other topics. Mr. Bravo was not only an excellent teacher—friendly, engaging, and full of memorable anecdotes—he was also perceptive and genuinely concerned that we will overcome our individual weaknesses. I still remember that a couple of months into the course, he deducted 30% of the points on an exam for a simple careless mistake, something he did not do with classmates who made similar errors. When I protested, he reminded me that I needed to learn to concentrate, because such mistakes were my Achilles’ heel. “I know how hard it is for you to accept this grade,” he said, “but one day you will thank me.” He was right. I never forgot it, and I have always remembered him with gratitude. He taught me that a good teacher cannot be governed by what a student thinks of him.
From that moment on, I followed his excellent advice on how to maximize our chances of success on exams:
- Read the entire exam before answering anything.
- If a question seemed unfamiliar, mark it and continue reading calmly.
- Answer first the questions we knew best.
- Avoid getting stuck too long on any one problem.
- Resist the temptation to turn in the exam immediately before reviewing everything carefully.
Answering the familiar questions first builds confidence. The mind works in parallel, and it was common for the solution to a difficult question to occur to us while working on another. Reviewing the exam also revealed how many simple, avoidable mistakes we made when we rushed.
I should clarify that when I finished university in Spain, I had never seen a multiple‑choice or true‑false question; all my exams were open‑ended. The technique I learned from Mr. Bravo helped me enormously throughout my studies in both Spain and the United States.
The last teacher I must mention is my doctoral advisor, Dr. Mark L. Teply. He was an excellent researcher who understood his field not as someone who had memorized it, but as someone who had truly mastered it, contributed new ideas, and kept up with ongoing developments. His lectures were somewhat monotone, but always remarkably clear, and he was unfailingly willing to help.
In 1975 I received a scholarship from the Catholic University of Puerto Rico, where I taught, to pursue doctoral studies. I had planned to study in Madrid, but when Franco fell ill that year and the political situation became unstable, I decided not to expose my wife and two‑year‑old daughter to unnecessary risk and instead went to the United States. Because of this last‑minute change, I arrived at the University of Florida with only a photocopy of my Spanish degree, without official transcripts, without having taken the GRE, and without having been formally admitted. I also spoke virtually no English—my wife had to interpret for me. Despite all this, Dr. Teply, who was the graduate advisor at the time, allowed me to enroll as a temporary student for one quarter. Two weeks later I took the GRE, which confirmed that I knew no English (10th percentile) but had a strong mathematical background (99th percentile). My grades that quarter dispelled any doubts about my potential, and I was soon officially admitted, granted tuition waivers, and later awarded a graduate assistantship.
Two years later I passed the Master’s and the doctoral qualifying exam. That summer I received a letter from my university in Puerto Rico stating that I needed to return the following year if I wished to keep my job. I never understood why, since it had been assumed that my studies would take the usual five years. Dr. Teply, who had agreed to supervise my dissertation, suggested that given the urgency, I should work in a new area of algebra—group and semigroup rings—where open problems were more accessible. He did not mind that it was not his specialty, even though it meant extra work for him. Nine months later he told me I had solved enough problems for a dissertation. To reassure me, he sent my work to a distinguished professor in ring theory, who replied that he would be delighted if one of his own students had produced such work. Thanks to this, I completed both the Master and the PhD in three years and kept my position in the Catholic University of Puerto Rico. I was told I was the first student in the history of the University of Florida to complete both degrees in mathematics in such a short time.
I do not believe I was the most intelligent student; some of my classmates were certainly brighter. But some of them never finished. I mention this because I sincerely believe that beyond faith and desperation —which can work miracles—there is a combination of ability and determination that matters more than intelligence alone. Time and again, in my classes with honor students, I have seen exceptionally intelligent students fail to find the time to do the work, while others with less talent but far more dedication always find the time and often finish earlier and better.
I have always been grateful to Dr. Teply. Without his vision and guidance, I could never have finished so quickly. Whenever I supervised a student doing research, I remembered and emulated what I learned from him:
- It is normal for a student with no research experience to feel lost at first. It is important to give them confidence that they can succeed.
- As they progress, we can help them refine their work and occasionally suggest new approaches or connections.
Graduate school in the United States—especially in STEM fields—offers advantages that are hard to find elsewhere in the Western world. Graduate students with strong records receive assistantships that include tuition waivers and stipends, and some universities provide low‑cost housing. Faculty conduct research regularly and are evaluated annually, so their salary increases depend partly on their productivity. In my experience, and that of many foreign colleagues, the treatment we received was excellent. I never felt discriminated against, and professors were always attentive both in class and in their offices.
Finally, during my graduate studies I had the opportunity to meet three of the great mathematicians of the twentieth century: Paul Erdős, Giancarlo Rota, and Stanislaw Ulam. They visited the department annually, interacted with faculty, and occasionally spoke with graduate students. It was extraordinary to hear Dr. Erdős—then one of the most prolific mathematicians in history—present open problems accessible to all of us. Dr. Rota, a professor at MIT, was a brilliant researcher who had written the most influential books on Functional Analysis and spoke five languages. Dr. Ulam had participated in the Manhattan Project and originated the Teller‑Ulam design for thermonuclear weapons. Their breadth of knowledge was astonishing, but what impressed me most was their humility. They made me realize that arrogance has no place in the world of knowledge and only reveals the ignorance of the one who displays it.
