Number theory
Number theory captivates with the simplicity of its statements and the unpredictability of its solutions. Number theory uses techniques from practically all mathematical divisions and there is almost one division of the number theory per mathematical divisions with gateways (or highways) enabling the transition from a division to another.
For example, the Fermat's Last Theorem (a cube is not the sum of two cubes, and more generally, a n-exponentiation is not the sum of two n-exponent), formulated around 1650, required for its solving the combined efforts of an impressive number of mathematicians for nearly four centuries. Its definitive solving in 1994, through the works of Ribet, Wiles and Taylor, run for more than two hundred pages of articles, which are themselves based on a few thoudand pages borrowed from various branches of mathematics (functions of a complex variable, representation theory, harmonic analysis, algebraic geometry...).
In the same vein, the problem of congruent numbers (which are integer that are the area of a right-angle triangle with sides of rational length) goes back to the 10th century, and it wasn't until 1983 that Tunneli gives a simple criterion for a number not to be congruent and, modulo the Birch and Swinnerton-Dyer conjecture (one of the Millennium Prize Problems), to make it congruent.
In a different genre, the distribution of prime numbers with all sorts od conjectures. The prime number theorem (giving an asymptotical formula for the number of prime numbers), foreseen by Euler, was demonstrated in 1896, using the holomorphic function theory and more specifically the properties of the Riemann zeta function in the complex plane. The Riemann hypothesis, formulated in 1858 (another Millennium Prize Problems), and with far-reaching consequences on the distribution of prime numbers, has resisted the repeated assaults of mathematicians up to now. Who knows where the solution will come from?
Green and Tao have demonstrated in 2004 that the set of prime numbers includes arithmetic progressions of arbitrary length, thus resolving a very old question. Their demonstration combines probabilistic ideas and other from the ergotic theory.
We conjecture (abc conjecture) that if a+b=c; where a, b, c are integer that are prime to one another, then c cannot be much larger than the product of the prime numbers dividing the product abc. A demonstration could be used to determine (in principle) the solutions in rational numbers of most equations in two variables.
The actual attacks are based on the Arakelov geometry, combining algebraic theory of classic numbers, algebraic geometry and fine analysis on varieties.
The problems above give an insight of the diversity of questions raisde, but does not cover the totality of the fields mentioned by the number theory (is missing, among other things, the transcendental number theory, algorithmical questions, applications to cryptography...).
Course language: French
- Teaching coordinator: Mauri Mirko