# Institute for Experimental Mathematics

A balanced mixture of basic and applied research representing the interdisciplinary perspective of the IEM offers excellent opportunities for rapid practical exploitation of the research results.

The activities of the Discrete Mathematics Research Group are rooted in a fundamental principle of mathematics, which says that a mathematical structure can be better understood by studying its symmetry group. Group theory methods – which are discrete by nature – can thus be used to study problems from geometry, algebra, number theory, topology, theory of functions and areas of application such as coding theory and cryptography.

The research group works on developing the necessary group theory tools and their application in other areas, much of which relies on the use of modern computer algebra systems like GAP, MAGMA and MAPLE.

Concrete research themes include highly symmetric algebraic curves and Riemann surfaces, invariant theory of binary forms, character and representation theory of finite groups, explicit constructions of presentations and representations, permutation groups with almost fixed point free elements, coding theory and cryptography.

The Digital Communications Research Group concentrates on the development of cryptography, information and communication theory. The founder of this discipline and the group’s role model is Claude Elwood Shannon (1916–2001). His 1948 paper “A Mathematical Theory of Communication” laid the foundation for the field of digital communications.

One of the main research topics of the group is attempting, like Shannon before them, to solve the problem of unreliable digital communication. They are working, for example, on the question of how a message, coded and transmitted by a sender but disrupted or corrupted in communication, can be decoded at the receiver end without any loss of data. This basic research is becoming increasingly important in today’s digitalized world.

In addition to the theory of digital communications, the research group has four other focal points: communication technology; multi-user communication and networking; coding; and data security. One long-running major research project on technical communication concerned the development of communication methods and coding techniques for Powerline Communications (PLC).

In a current project relating to data security, new public key systems are being investigated and developed on the basis of factorization of finite groups. In contrast to the public key systems used today, the security of such a well designed system would also be guaranteed for the coming generation of quantum computers.

Another project is currently investigating the principles of safe transmission of biometric data. Whether fingerprints or facial physiognomy, biometric data change and thus make it more difficult for security systems to identify them with absolute reliability. Safe storage of biometric data is another important issue for society. Optimizing security systems and protecting critical infrastructures such as energy, telephone and transport networks are other tasks which involve the work of data transmission experts.

The Alfried Krupp von Bohlen und Halbach Chair for Computer Networking Technology has two focus areas in its research: new network technologies, architectures and their protocols; and current network security issues. In the area of internet protocols, the group conducted successful research into the systematic evaluation and evolution of the transport protocol SCTP in a joint project with the University of Applied Sciences Münster funded by the German Research Foundation (DFG). In particular, the work on the concurrent use of multiple network paths (Concurrent Multipath Transfer) yielded important results and led to important publications.

In a project funded by the Federal Ministry of Education and Research (BMBF), the group’s two competence areas were 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.

Detecting attempted fraud and misuse in IP- based telephony (Voice over IP) and the development of suitable mitigation mechanisms are other areas in which substantial achievements were made. Among them is the launch of a new BMBF-funded project in which the group is working with Fraunhofer Fokus and several SMEs.

Research on the security of peer-to-peer networks is another focus topic. The research activities of the group are thus contributing to making the current and the future internet better and more secure for the various multimedia applications. An explicit goal of the Chair – in addition to international publication – is to contribute its research results directly to the relevant standardization to ensure their practical application on a global scale. The research area of the Number Theory Research Group is arithmetic geometry and algebraic number theory.

The fundamental concern of algebraic geometry is how to understand the structure of the solution set of polynomial equations geometrically. A simple example is the equation y^{2}=x^{3}+Ax+B, where A and B are integers, which typically define elliptic curves. A sound understanding of elliptic curves and their parameter spaces is of crucial significance not only in theoretical number theory but also for applications in cryptography.

Rather than curves, it is also possible to examine analogous varieties of higher dimension known as Abelian varieties. One compelling aspect here is that modern algebraic geometry permits geometric intuition to also be applied to problems of number theory. One goal is to prove the conjectures recorded in the Langlands program regarding the symmetry of the roots of polynomials in one indeterminate with integral coefficients. While this is a classical topic with interesting results dating back hundreds of years, such as the proposition that there cannot exist a general formula (as that for quadratic equations) for the solutions of polynomials of degree greater or equal to 5, major progress has also been made in this field over the past years. However, numerous questions still remain unanswered.

The Number Theory Research Group is engaged in researching current theoretical problems and in the extensive application of explicit, algorithmic and experimental methods: deep insight is often gained through knowledge of paradigms that can be calculated exclusively on the computer; at the same time, a sound understanding of abstract correlations often proves to be extremely fruitful or even indispensable in performing heretofore impossible calculations and opening up novel potential applications.