• For everybody: Mathematics, algebra, algebraic geometry, number theory, in increasing degree of accuracy. (To find out what number theory is about, read on.)

  A short description of Number Theory:

  • Many people find whole numbers interesting. Some special numbers are thought to be more relevant than others, for instance 7 is often regarded as "lucky number". The taxicab number 1729 can be written as the sum of two cubes in two different ways:
    1729=10³+ 9³ =12³+1³ ,
    and it is the smallest number with this property.
    (The plate number of the cab which Hardy took to visit Ramanujan in a hospital was 1729, which Hardy thought was a "boring number". When told of this number, Ramanujan's reply was "No, Hardy, it is a very interesting number, because ...")
    
  • Many applications of number theory has been discovered in recent years, including coding theory, graph theory and cryptology. With computers invading all aspects of modern life, number theory has gained more attention in the "real world"; e.g. if you are the unlucky inventor of a fast way of factoring integers, be sure to get a very good lawyer. No bluffing —the National Security Agency is funding research in number theory. (End of my sales pitch.)
    
  • One typical problem in number theory is to solve equations in integers. The most famous example, know as Fermat's Last Theorem and recently proved by A. Wiles, asserts that for any integer n > 2, it is not possible to find nonzero integers x, y, z such that
    xⁿ + yⁿ = zⁿ .
    (Although the statement of Fermat's last theorem is fairly elementary, it is unlikely that an elementary proof will be found. Instead Wiles's proof uses sophisticated tools developed in the last several decades.)
  • Another influential problem in number theory is to understand how prime numbers are distributed among natural numbers. Here are two things we know about the prime numbers. Among the first N natural numbers, roughly N/ln(N) are prime numbers; furthermore among these prime numbers, when divided by 4, about half of them have remainder 1 and about half of them have remainder 3. (This is the statement of the prime number theorem.)
  • Here are two seemingly unrelated facts.
    
    • The number
      e^{\pi*\sqrt{163}} = 262537412640768743.9999999999992... is very close to an integer—the difference is less than 10^-12. (Most hand-held calculator will declare that this is an integer; in fact it is a transcendental number.)
      
    • The polynomial x² - x + 41 takes prime values for x = 0, 1,...40.
    Both of the above statements are explained by the fact that the quadratic number field with discriminant -163 has class number 1 .
  If this kind of stuff interests you, consider taking a course in number theory. You will learn a lot about number theory and some "hot" topics, including some public key cryptography and quantum computing. To read a few more pages about this, I have some short notes on elementary number theory.

• Geek talk: Arithmetic Algebraic Geometry. More specifically:
  • Abelian varieties and their moduli
  • p-divisible groups and their l-adic Galois representations
  • Crystals and Dieudonné Theory
  • Shimura varieties, their models and reduction
  • Motives and L-functions
  • Exponential sums
  • Hodge theory
  • Algebraic groups and arithmetic subgroups

Papers and Preprints.

Acknowledgement: Ching-Li Chai's research was supported by the National Science Foundation over the years, including a three-year grant DMS01-00441 and a five-year grant DMS04-00482.

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