ABSTRACT Hendrik Lenstra Berkeley 9732709 Professor Lenstra will study a collection of problems in algorithmetic number theory, algebraic number theory, and algebraic geometry. Topics to be studied include smooth numbers, Mersenne primes, Galois groups of local fields, Belyi functions, Lacunary polynomials, and polynomials over finite fields. Number theory is one of the oldest branches of mathematics. It is the study of the natural numbers, the basic numbers of counting. From the very beginning of the area, there has been a gap between what was known to be possible for all numbers, and what could actually be done for a specific one. For example, it is known that any natural number can be factored in only one way as the product of smaller unfactorable numbers. Yet given a carefully chosen large number, it can be so difficult to factor into parts that scientists have developed secure codes based on factorization. This project is devoted to improving the methods we have for actually computing some of the most important mathematical properties of particular numbers. In the computer age, the potential applications for improved methods of computation go well beyond mathematics and science and into everyday life.
