Only forty eight people in history have ever been awarded the Fields Medal the highest honor a mathematician can receive and Enrico Bombieri is one of them.
The Fields Medal Award was created in 1936 by the Canadian mathematician, John Charles Fields. It is awarded every four years to either two, three, or four outstanding mathematicians who have made significant contributions to important areas of mathematics.
Bombieri was awarded the medal mainly for his contributions to prime sieve theory and the study of multidimensional surfaces. But he is also known for his work in other areas, including univalent functions, partial differential equations involving minimal surfaces, the Bieberbach conjecture, and other areas of number theory.
Enrico Bombieri was born in Milan, Italy in 1940 and the mathematics bug struck him early in life. By the time he was thirteen he could be found pouring over a textbook on number theory, fascinated by its order and purity. Later he went to the University of Milan and studied with G. Ricci, then moved to Trinity College in Cambridge where he studied with the eminent number theorist, Harold Davenport, author of the classic, The Higher Arithmetic.
After receiving his Ph.D. in 1963 from the University of Milan, he went on to teach at the University of Pisa and then the Scuola Normale Superiore. During this time he developed a reputation for being “the Mathematical Aristocrat” since he would regularly show up for math conferences driving a fast, expensive sports car. Once, a rumor even surfaced that he had placed sixth in an Italian car race. Also, many of his colleagues knew he came from a wealthy Italian family and that his father was an economist who owned several vineyards. Enrico’s hobbies of painting and collecting wild mushrooms were also well-known.
In 1974, he left Italy for the United States where he took a position at the Institute for Advanced Study in Princeton, NJ. Today he still holds the IBM von Neumann Professorship of Mathematics there. The Advanced Institute specializes in cutting-edge research in many different scientific areas and Bombieri is considered one of the world’s experts on analysis and number theory.
Shortly after arriving at the Advanced Institute, he was awarded the Fields Medal for his contributions to multidimensional surfaces and prime sieve theory, as previously mentioned. To be considered for the Fields Medal, mathematicians must be under the age of forty to qualify. A monetary sum is given, which in U.S. dollars amounts to approximately $13,500. The Fields Medal is considered the equivalent of the Nobel Prize in Mathematics (even though the money received is significantly less than the Nobel award).
Outside of the work he did to win the Fields Medal, perhaps Bombieri’s most notable achievement is his improvement upon Linnik’s “large sieve” method, which helped to demonstrate a result now known as “Bombieri’s mean value theorem.” Simply put, the theorem involves understanding the way prime numbers are distributed in arithmetic progressions.
Let’s digress for a moment and recall that a prime number is a positive integer divisible only by itself and one. For example, 6 is not a prime because it has the divisors 1, 2, 3, and 6. But 11 is prime because its only divisors are 1 and 11. The sequence of primes begins 2, 3, 5, 7, 11, 13, 17, 19, … and theorems concerning them are among the most beautiful in mathematics. Euclid proved in ancient times that there are infinitely many prime numbers and his proof still holds up as one of the clearest and most insightful to this day. Euclid proved we will never run out of primes but only continue to find more. For example, the currently largest known prime number is 2^30402457-1 found by Dr. Steven Boone and Dr. Curtis Cooper, both professors at Central Missouri State University. Their prime consists of 9,152,052 digits, yet many other people around the world at this moment are looking for a larger prime.
Perhaps my favorite sequence concerning prime numbers is the repunit primes. Repunits are simply copies of the digit 1 repeated: 1, 11, 111, 1111, 11111, … and they have the formula, r(n) = (10^n1)/9. The only known values of n for which r(n) is prime are 2, 19, 23, 317, 1031, 49081, and 86453. To illustrate the intriguing and malleable nature of primes, we will create our own function involving r(n) to show how easy it is to find more of them. Letting z(n) = n * r(n) * 10^n + 1, (which has an unusual decimal expansion), z(n) is prime for n = 1, 5, 44, 56, 187, 192, 206, and there are no more up to n = 4,000.
But let’s return to Bombieri.
He also spent some time working on the Riemann Hypothesis, which is currently regarded as the “Holy Grail” of mathematics. Anyone who solves it will immediately achieve immortality. Bombieri first read about the Riemann Hypothesis at the age of fifteen. Even though the problem is too advanced to fully explain here, suffice it to say that it concerns summing the infinite series: 1 + 1/2^s + 1/3^s + 1/4^s + … where s is a complex variable. There is a million dollar award offered by the Clay Institute to any mathematical leviathan who succeeds in defeating the Riemann Hypothesis. But beware that many geniuses have already fallen short of the task. Even though Bombieri has worked on this problem for many years and has not yet succeeded in proving it, he is not bothered because he says he is more concerned with finding the beautiful results that arise along the way, in addition to the pleasure received from working with the natural beauty and harmony inherent in number theory.
Colleages commenting on Bombieri have said he possesses incredible insight into mathematical structure, so much so that he can enter a new field far from his original comfort zone and master the complicated essentials with ease and efficiency; then pick just the right problems to attack using deep results from other areas no one has ever thought of employing. Also Bombieri is known as a lucid writer and explicator of mathematics, and also well-known for delivering lectures with character and clarity.
Bombieri has been awarded a number of prizes and honors throughout his career in addition to the Fields Medal. In 1980 he received the Balzan Internation Prize and in 1984 he was inducted into the French Academy of Sciences. He has also been awarded the Feltrinelli Prize and the Balzan Prize, while also joining the American Academy of Arts and Sciences; the Royal Swedish Academy; Rome’s Accademia Nazionale dei Quaranta; the National Academy of Sciences; and the European Academy of Sciences, Arts, and Humanities, among others.