1.1. Number TheoryProgramme leaders: H.W. Lenstra, R. Tijdeman
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.
Status of the programme.
The number theorists of the Stieltjes Institute made important contributions to the current development and continue their active participation. Their expertise and interests cover most of presentday number theory.
There is an excellent research climate for number theory both within the Stieltjes Institute and at a national level, partly as a result of the "groot project getaltheorie" of NWO/SWON, in which the universities of Leiden, Amsterdam, Groningen, and Utrecht participate. The very successful biweekly Number Theory Seminar, which is organized from within the Stieltjes Institute, draws large audiences and serves as a meeting ground for number theorists and as an introduction to research for graduate students. There is a cooperation project between Leiden, CWI Amsterdam, and Groningen on the factorization of integers. In addition there is a regular flow of visitors and postdocs.
In April 1999 there will be a workshop in number theory at the Lorentz Centre in Leiden. Amsterdam will, in July 2000, be the venue of the fourth Algorithmic Number Theory Symposium (ANTS), one of the major events worldwide in computational number theory.
On an international level, there have always been close contacts between number theorists, through correspondence, meetings, and personal visits. These have further intensified as the result of the development of modern means of communication and the demise of the Soviet empire. The number theorists in Stieltjes have close and longstanding cooperations with mathematicians in virtually all countries where number theory is done, including the United States, Canada, England, France, Germany, Italy, Hungary, Poland, India, Japan and Australia.
Research Staff (situation at January 1,2007)
 Permanent staff
 Dr. J. Brinkhuis (EUR) (from programme 4.1)
 Prof.dr. R.J.F. Cramer (UL/CWI)
 Dr. J.H. Evertse (UL)
 Prof.dr. J.P. Hogendijk (UL/UU)
 Dr. C. Kraaikamp (TUD) (from programme 3.1)
 Prof.dr. H.W. Lenstra (UL)
 Dr. B. de Smit (UL)
 Prof.dr. P. Stevenhagen (UL)
 Dr. R.J. Stroeker (EUR)
 Prof.dr. R. Tijdeman (UL)
 Dr. R.W. van der Waall (UvA)
 Ph.D. students
 Drs. J. Bouw (UL)
 Drs. J. Brakenhoff (UL)
 Drs. J.L.A.H. Daems (UL)
 Drs W.H. Ekkelkamp (UL/CWINWO)
 Drs. R. de Haan (UL/CWI)
 Drs. B.J.H. Jansen (UL)
 Drs. W.J. Palenstijn (ULNWO)
 Drs. S.W. Rosema (UL)
 Ir. I. Smeets (UL)
 Drs. T.C. Streng (UL)
 E.L. Toreao Dassen, MSc. (ULEU)
 CWI participants
 Dr. S. Fehr
 Dr.ir. H.J.J. te Riele
