# Institute for Experimental Mathematics

A balanced mix of basic and applied research in line with the interdisciplinary concept behind the IEM offers excellent opportunities for quick utilisation of research results.

The research activities of the Discrete Mathematics Group are rooted in a fundamental mathematical principle, according to which a mathematical structure can be understood better by studying its symmetry group. Methods of group theory - which are discrete - can thus be used to study problems from geometry, algebra, number theory, topology and theory of function, and in various areas of application such as coding theory and cryptography.

The team works on providing the necessary group theory tools and their application in other disciplines, much of which depends heavily on the use of modern computer algebra systems such as GAP, MAGMA and MAPLE.

Current research themes include highly symmetric algebraic curves and Riemann surfaces, invariant theory of binary forms, the inverse problem of Galois theory, character and representation theory of finite groups, permutation groups with almost fixed point free elements, coding theory and cryptography. A research project is currently being prepared in which the explicit forms of certain invariants are to be collected in a database for various applications.

The Digital Communications Group concentrates on the mathematical theory of communication. The founder of this discipline and the group's role model is Claude Elwood Shannon (1916-2001), whose 1948 standard work "A Mathematical Theory of Communication" laid the mathematical cornerstone of digital communications.

One of the group's main research concerns, like Shannon before them, is solving the problem of unreliable digital communication. They are looking into the question of how to reconstruct a transmitted message which has been disrupted or corrupted during transmission, at the receiver end without any loss of information. Such basic research is becoming increasingly important in today's digitalised world.

In addition to the theory of digital communication, the research group has four other focal points: communication technology, multi-user communication and networking, coding, and data security. One of the major long-term research projects in communication technology concerned the development of communication methods and coding techniques for power line communications (PLC).

An ongoing project on the subject of data security is investigating and developing new public key systems based on the factorisation of finite groups. In contrast to the public key systems used today, the security of such a well-designed new system would also be guaranteed for the coming generation of quantum computers.

Another current project is looking into the safe transmission of biometric data. Whether they are collected from fingerprints or facial physiognomy, biometric data can change, making failsafe verification by security systems difficult. Safe storage of biometric data is another subject of major importance to society.

Optimising security systems and protecting critical infrastructures such as energy, telephone or transport networks are other areas which always involve tasks for data transmission specialists.

The Alfried Krupp von Bohlen und Halbach Foundation Chair for Computer Networking Technology has two areas of focus in its research: new network technologies, architectures and their protocols, and current network security issues. Research activities related to a novel IETF (Internet Engineering Task Force) concept for Reliable Server Pooling were successfully completed together with the corresponding project funded by the German Research Foundation (DFG). The PhD thesis written by Thomas Dreibholz in connection with the project was recognised with the Research Award of the Sparkasse Essen, while Pascal Schöttle's Bachelor dissertation also relating to the work received the CAST award for IT security. Contributions and active editorial participation made a significant contribution to final approval of the corresponding IETF standards at the end of 2008. Research on the development of the new internet transport protocol SCTP (Stream Control Transmission Protocol) is continuing in cooperation with the University of Applied Sciences Münster in a joint DFG project. In a project funded by the German Ministry of Education and Research (BMBF), the two competence areas of the group are being combined and further developed in cooperation with Fraunhofer Fokus and the TU Kaiserslautern. The consortium focuses its research on basic architectural and security issues for the "Future Internet" within the German research initiative G-Lab. The security of peer-to-peer networks and the protection of IP-based telephony (Voice over IP) are other current research topics. In this way, the research activities of the group are contributing to improving the usability and security of the present and future internet for the many voice and multimedia applications. One of the group's explicit aims, in addition to international publication of its research findings in the scientific press, is to directly incorporate them in the relevant standardisation to ensure their practical application on a global scale.

The core of the activities of the Number Theory Group is research in arithmetic geometry and algebraic number theory. A large proportion of the research questions are related to the Langlands programme, a dominant and very topical subject in number theory. The principle concern of algebraic geometry is to obtain a geometric understanding of the structure of the solution set of polynomial equations. A simple example is x²+y²=1, whose solution set (the real plane) is the circle of all points with a distance of 1 from the origin. Generally speaking, describing the geometric properties of a given system of equations is very complicated. A particularly interesting aspect of this problem is that geometric intuition can be applied with this theory to questions of number theory.

There are close connections between the current theoretical research of the Number Theory Group and the broad application of explicit, algorithmic and experimental methods: deep insights are often founded on knowledge of examples that can only be obtained from computer calculations; at the same time, profound theoretical knowledge often proves beneficial or even indispensable for performing otherwise impossible computations and discovering new applications. Data security, which comprises cryptography and coding theory, is a particular area in which applications exist.