Algebraic geometry

Algebraic geometry studies geometric objects – curves, surfaces and their high-dimensional relatives – that are given by polynomial equations. The subject unites algebra and geometry in a way that allows abstract concepts to have concrete geometric meaning. The course takes its starting point in algebraic curves of low degree, where we can use relatively elementary methods to explore phenomena such as tangent lines, singularities and addition laws on cubic curves. These examples demonstrate both the strength and limitations of the classical methods and justify the need for more systematic algebraic tools, which we then also introduce. The course thus provides a first insight into modern algebraic geometry, with classical examples as guidance.