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1.1 Number Theory

Programme leaders: H.W. Lenstra, R. Tijdeman

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, theoretical computer science, and dynamical systems theory. These new developments stimulated the design, analysis and use of algorithms, now called computational number theory.
The research in the Number Theory programme ranges from algebraic number theory and group theory to analytic number theory and ergodic theory. A central theme is the design and application of algorithms in number theory. In algebraic number theory the focus is on algebraic units, class groups and Galois module structures. Methods from class field theory, homological algebra and analytic number theory play an important role. Methods from algebraic number theory and numerical mathematics are applied to factorization methods for large integers. Computational methods from algebraic number theory and diophantine approximation are applied to determine all integer or even rational points on algebraic curves, and, besides ineffective methods, to study diophantine equations and recurrence sequences. Probabilistic and combinatorial aspects of continued fraction expansions are studied. This is interrelated with the theory on combinatorics of words. In group theory the research deals with representation theory of groups. The long list of publications reflects the activities of the project group.

The biweekly national Intercity Number Theory Seminar continued to be the meeting place for the participants. Besides there were several successful activities supported by the Spinoza grant of H.W. Lenstra and the Stieltjes Institute. The appointment of P. Stevenhagen to Professor at Leiden University confirmed the strong position of Leiden as international centre of number theory.
The Algorithmic Number Theory Symposium ANTS4 in Leiden in July 2000 attracted an international crowd of 140 people. It was preceded by a Number Theory workshop on a smaller scale in the Lorentz Center in June 2000.
In the same week the manifestation Pi in de Pieterskerk attracted national attention and turned out to be an excellent presentation of mathematical science to the public.
The Lorentz Center also hosted the Kloosterman Centennial Celebration in April 2000, which celebrated the 100th anniversary of the Leiden professor H. D. Kloosterman (1900 - 1968).

There were three well attended Stieltjes Weeks in the Lorentz Center. These workshops are a meeting place for Ph.D.-students and staff and provide a broad introduction to an important research field. The topics were Riemann surfaces (Stevenhagen, de Smit, Wiegerinck, Opdam), L-functions from algebraic geometry (H.W. Lenstra, P. Stevenhagen), and Algorithms in number theory (H.W. Lenstra Jr., P. Stevenhagen, H. van Tilborg) in collaboration with EIDMA.
Prof. M. Zieve (University Southern California, U.S.A.) visited Leiden during the spring semester and gave a series of lectures in the Intercity Number Theory seminar on curves over finite fields.
The Stieltjes Prize 1999 was awarded to Nils Bruin for his thesis. He also received the Kok Prize of the University of Leiden.

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Next: 1.2. Geometry Up: Algebra and Geometry Previous: Algebra and Geometry