Algebraic curves

Algebraic curves

4 hours of lectures per week.

Lecturers:

Johan P. Hansen and NN.

Content:

An algebraic curve is the zero set C = (x , y) P (x , y) = 0 of a polynomial P (x , y) k x , y,
where k is a field like for example the complex numbers. The theory of algebraic curves was
founded in the previous century and is a rich source of beautiful mathematics. For example if
P(x,y) = y² - x³ - x , then the curve C is an abelian group(!!) where addition is defined
geometrically using chords and tangents. Surprisingly curves like this one appear naturally in the
proof of Fermat's last theorem. Using only basic algebra this course is a mild introduction
(leading to more advanced algebraic geometry) to the elementary methods of algebraic curves
centered around projects like

Algebraic curves and Fermat's last theorem

Algorithms for prime factorization using algebraic curves

Riemann-Roch's theorem

Parametrization of curves.

Prerequisites:

The course assumes knowledge of Algebra 1. Participants must be prepared to meet a short
introduction to a powerful computational method for solving equations. The beauty and
simplicity of this method has been known to make women swoon and grown men cry.

Literature:

W. Fulton: "Algebraic Curves", Addison-Wesley.