Algebraic Geometry

Algebraic geometry seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mathematics. In one respect this last point is accurate. – David Mumford

Algebraic geometry is one of the oldest and at the same time one of the most active research areas in mathematics. To put it simply, Algebraic Geometry studies of geometric spaces, that are described by particularly simple equations, but can exhibit very complicated geometry. The field fascinates many mathematicians because it combines geometric intuition with highly abstract concepts of modern algebra and number theory.

Algebraic Geometry has many Links to other areas of mathematics, such as the Number Theory, Topology, Representation Theory and the Complex Analysis. It plays a role in some areas of Theoretical Physics plays and is an indispensable tool for modern data security and encryption technology.

Beispiel: Diagonalfläche von Clebsch

The picture below shows the “Clebsch Diagonal Surface”. This is a surface \(S\) that is given as a zero set of an equation of degree three. The mathematician uses “projective coordinates” for a technically correct definition and writes

$$S = \Bigl\{ [x:y:z:w] \in \mathbb P^3 : (x+y+z+w)^3 = x^3 + y^3 +z^3+w^3 \Bigr\}.$$

It has been known since the middle of the 19th century that every surface that can be described by an equation of degree three contains exactly 27 straight lines. Three of these degrees are drawn in the picture. Do you see the missing 24 straight lines? The symmetry of the surface can help you!

Visitors of the Mathematical Institute of the Albert-Ludwigs-University of Freiburg will find a historical plaster model of Clebsch’ Diagonal Surface in the corridor on the 3rd floor, on which all 27 degrees are drawn in.

The page Imaginary of the Oberwolfach Mathematical Research Institute contains very nice pictures and many interactive programs that illustrate mathematical facts and concepts.