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algebraic geometry

Algebraic Geometry

After the invention of coordinates by Descartes it soon became clear, that important geometric objects (such as lines, conic sections or the lemniscate) could be described by polynomial equations. The sphere, for instance, is the solution-set of the equation

$x^2+y^2+z^2 = r^2.$

The variables x,y,z denote the coordinates in space, the parameter r is the radius of the sphere under consideration.

Algebraic geometry aims at systematically investigating the geometry of solution-sets of systems of polynomial equations.

Of course this basic problem has been extended in several directions. To any commutative ring there can be associated a geometrical object (its spectrum), and the objects of modern algebraic geometry (schemes)
are modelled using those spectra as basic building blocks. On the other hand, the geometry of complex spaces (with holomorphic mappings) is to a large extent very parallel to the algebraic world.

The following main branches of algebraic geometry are to be found at our work group:

Classification of varieties

The simplest case of this is to classify the solution-sets of polynomial equations, which have dimension 1, i.e. algebraic curves. If the parameters of the equations are varied "continuously", the solution-curve is deformed in some way. It a the main problem then firstly to make this vague notion precise and secondly how to work concretely with such a moduli space, the points of which correspond to isomorphism classes of of solution sets with some common restrictions. To that end, the group of Prof. Herrlich studies the geometry of moduli spaces.

Arithmetic geometry

Here one studies the solution sets of polynomial equations over some fixed (commutative) ring, mainly over the ring of rational integers. This is closely related to number theory, and L-series as well as Galois-representations play an important role.

Applied algebraic geometry

is usefull in coding theory, cryptography, and robotics. In Karlsruhe, this is mainly studied at the faculty of computer sciences.