1.1. Number Theory
Programme 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.
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