After successful completion of the course, students are able to describe and apply fundamental concepts and methods of the theory of algebraic curves.

This theory has connections to the number theory, projective geometry, and complex analysis. In particular we will learn how to find all integer solutions of the equation x^2 - 21y^2 = 1, give algebraic proofs of incidence theorems of projective geometry, and use the complex structure in order to derive further geometric theorems.

Conics: affine and projective classification, rational parametrization, integer points on hyperbolas, pencils of conics.

Quadrics: classification, pencils, geometric theorems.

Cubics: group structure, parametrization with elliptic functions.

Lecture, discussion of examples, homework.

The lecture will be given via Zoom, the communication done through TUWEL. Please register in TISS, the registration will be transferred to TUWEL.

Homework, oral exam. Possibly visualization in Mathematika or Geogebra. Projects for school.

Stillwell "Numbers and geometry"

Gibson "Elementary geometry of algebraic curves"

Reid "Undergraduate algebraic geometry"

Matrix calculus, complex numbers.