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Research Programme 1.1: Number Theory and Algebra

Programme leader: H.W. Lenstra

Staff

(situation at November 1, 2008)

permanent staff

emeritus

PhD students

guest researchers

Description of the project

The main focus of the research programme is number theory. Number theory studies the properties of integers, with a historically strong emphasis on the study of diophantine equa-tions, 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, and in recent times algebraic geometry has become increasingly im-portant. Another recent development is the discovery that number theory has significant im-plications in more applied areas, such as cryptography, theoretical computer science, the the-ory of dynamical systems, and numerical mathematics. This discovery led to the rise of algo-rithmic and computational number theory, which occupies itself with the design, analysis, and efficient implementation of arithmetical algorithms. The overall result has been a unification rather than a diversification of number theory. For example, the applications in cryptography depend heavily on algebraic geometry, and algebraic number theory, which used to stand on itself, is now pervading virtually all of number theory. Themes of the programme reflect the research areas mentioned. They include finding points on algebraic curves, applications of group theory and algebraic number theory, the theory of finite fields, diophantine approxima-tion, words and sequences, discrete tomography, primality tests and factorization methods, and the development of efficient computer algorithms. The algebra portion of the programme is strongly oriented towards the applications of algebra in number theory and arithmetic geometry and towards algorithmic aspects. Themes include Galois theory and various aspects of group theory and ring theory. The research programme also includes cryptology and the history of mathematics. Main themes in cryptology are the applications of number theory and algebra to the design of cryp-tographic schemes, and foundational issues are considered as well. In the history of mathe-matics, the emphasis is on the edition and translation of early Islamic mathematical and as-tronomical texts.