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for Ngô Bao Châu |
The awarding of the Field Prize in Mathematics to Ngô Bao Châu was more than the well-deserved honoring of a young mathematician; it was also a mile stone for the Vietnamese people and culture. There probably had been comparable geniuses in previous generations but those people of genius never had the opportunity to utilize their spectial talents. A society on the edge of sarvation cannot afford to let some indulge in esoteric pursuits like mathematics. And even if such geniuses had had free time to study the Vietnam of the past did not have the books required to learn the advanced mathematics for research into even more advanced mathematics. Thus Ngô Bao Châu's achievement reflects the improvement in education and social opportunity of modern Vietnam. Vietnamese people are justifiably proud of Ngô Bao Châu's achievements and understand that his honor is also, in some small part, their honor.
Ngô Bao Châu was born in 1972 into an academic family in Hanoi. His father was a physicist and his mother a professor of medicine. Bao Châu was their only son. At age 15 he was admitted to special courses in mathematics at the high school in Hanoi preparing students for the National University. In the final two years of high school he entered the International Mathematical Olympiads and won gold medals both times.
Bao Châu was planning to go to Budapest on a Hungarian government scholarship, but the downfall of the communist government resulted in the withdrawal of that scholarship offer. What appeared to be a setback turned out to be fortuitous because he received an offer of a scholarship to study mathemetics in France. He went to France and studied at the Ecole Normale Supérieure. He stayed on in France an ultimately received a doctorate of philosophy in mathematics in 1997 from the Universite Paris-Sud. He continued to study mathematics at the Paris 13 University during the period 1998 to 2005. In 2004 he and the French mathematician, Gérard Laumon, under whom Ngô Bao Châu sudied at the Universite Paris-Sud were award the Clay Research Award. In 2005 he became a professor of mathemtics at the Universite Paris-Sud 11. In that same year he was awarded the title of professor in Vietnam. He later became a professor at the Hanoi Institute of Mathematics and took an appointment to the Institute of Advanced Study at Princeton University in New Jersey. More recently he accepted an appointment to the University of Chicago.
The mathematical research of Ngô Bao Châu and his mentor, Gérard Laumon, had do with a research program outlined by Robert Langlands. This program had to do with mathematical groups at a very high level. A mathematical group is a set of elements and an operation analogous to addition or multiplication. Reduced to basics an operation is just a function of two variables. The set and operation of a group have to satisfy certain axioms (requirements). One of these requirements is that the operation be associative; i.e., (a+b)+c is equal to a+(b+c). A group operation does not, however, have to be commutative; i.e, a+b can be different from b+a. There has to be an identity; i.e, an element e such that for for all elements of the group a+e is equal to a. There has to be an inverse for every element; i.e., for any a there is a b such that a+b is equal to the identity e. Usually the inverse of a is denoted as (−a) or a^{-1}.
The theory of groups is surprisingly rich. The young French mathematician Evariste Galois formulated it to establish that there is no general formula for solving fifth degree polynomial equations.
Ordinary numbers and addition form a group. Integers and remainder arithmetic also form groups. For example, suppose numbers are added together and the remainder is taking with respect of division by nine. Thus 8+7=6 in this arithmetic because the remainder for 15 upon division by 9 is 6. The elements of the set are {0, 1, 2, …, 8}. The group identity is zero and the inverse of each integer n is (9-n).
An interesting application of this nine's remainder arithmetic is digit sum arithmetic. The digit sum of a number is obtained by adding its digits together until a single digit is obtained. The digit sum of 125 is 8. The digit sum of 88 is 7 because 8+8=16 and 1+6=7. One interesting thing is that the digit sum of the product of two numbers is the product of their digit sums. For example, 125*88 is 11000. The digit sum of 11000 is 2. The digit sum of 125 is 8 and that of 88 is 7. The product of 8 and 7 is 56 and the digit sum of 56 is 2 because 5+6=11 and 1+1=2.
In a group there will be subsets of the elements which constitute a group. These are called subgroups or factor groups of the group.
This all is of course baby mathematics compared to the mathematics Langland, Laumon and Ngô Bao Châu were dealing with, but it introduces the concept of groups and their properties. One rich line of development involves groups whose elements are matrices. The matrices can be arrays of real numbers but they also can be arrays of complex numbers. When certain restrictions are imposed upon these matrices geometric properties are introduced along with the algebraic properties. This results in very interesting interplay of geometric and algebraic considerations. For example see Lie Algebras and Subatomic Particles.
(To be continued.)
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