1.1. Number TheoryProgramme leaders: R.J.F. Cramer, H.W. Lenstra
Description of the programme.
Number theory studies the properties of integers, with a historically strong emphasis on the study of diophantine equations, that is, systems of equations that are to be solved in integers. The methods of number theory are taken from several other branches of mathematics. Traditionally, these include algebra and analysis, but in recent times algebraic geometry has been playing a role of increasing importance as well. It has also been discovered that number theory has important applications in more applied areas, such as cryptography, dynamical systems theory and numerical mathematics.
These new developments stimulated the design, analysis and use of algorithms, now called computational number theory. They led to a unification rather than diversification of number theory. For example, the applications in cryptography are strongly connected to algebraic geometry and computational number theory; and algebraic number theory, which used to stand on itself, is now pervading virtually all of number theory.
Themes of the programme reflect the mentioned research areas. They include finding points on algebraic curves, applications of group theory and algebraic number theory, the theory of finite fields, diophantine approximation, almost periodic and recurrence sequences, primality tests and factorization methods, and the development of efficient computer algorithms.