# Research

## ALGORITHMIC ARITHMETIC GEOMETRY

The research group `Algorithmic Arithmetic Geometry' studies number-theoretic (or `arithmetic') properties of algebraic varieties like algebraic curves or surfaces that are defined over the rational numbers (or more general algebraic number fields). The set of rational points on such a variety is an important arithmetic invariant, and it is an interesting problem to describe its structure. In more down-to-earth terms, we are looking at the set of solutions in rational numbers of a system of algebraic equations. If we have one equation in two variables, then the solutions in question are the rational points on a curve. There are theoretical results that imply that the set of rational points on a curve always has a finite description. We are interested in the algorithmic aspects: How can we actually find this finite description for a given curve? Is this always possible? And what happens if we consider higher-dimensional varieties instead of curves?

The advanced seminar “Arithmetic Geometry” takes place weekly during the lecture period.

## CODING THEORY

The research group `Coding Theory' studies linear codes using methods from combinatorics, algebra, finite geometry and combinatorial optimization. A linear code is a subvectorspace or a submodule of a finite ambient space. One of the most interesting question comes from the application: how good is such a code if we fix the size of code and the size of the ambient space. How big is minimum distance or equivalently how many errors can be corrected if we use this code for the transmission of messages. In many cases we were able to construct the currently best known codes. The next step is to show that such a code is optimal, meaning that it is impossible to construct a code with higher minimum distance.

A new aspect is coding theory for networks (like the internet). Here we are faced with new questions, as the communication which we want to secure against errors using coding theory is no longer a point to point communication, as we may have many senders and many receivers (like TV-streaming, distributed storage).