Discrete mathematics is a rather young mathematical field that has experienced explosive growth in the last forty years. This development has several clearly discernible components:
• A substantial body of theory has been developed for many areas of discrete mathematics, with important subfields such as combinatorial enumeration, graph theory, discrete geometry, and the theory of polytopes.
• Connections to other fields of mathematics have enriched discrete mathematics by a multitude of methods (algebraic, topological, geometric, probabilistic, etc.).
• Combinatorial methods have found substantial use and applications in practically all modern parts of mathematics. They lie at the basis of computer science, not only in the theory of algorithms and complexity theory, but also in application fields such as computational geometry, computer aided design, and visualization.
• Combinatorial optimization (both in its theory and in applications) heavily draws on a wide repertoire of discrete mathematics tools.

The area of combinatorial optimization is indeed a place where very visibly discrete mathematical concepts have led to substantial impact not only on the theory of optimization, but also in genuine industry applications. This concerns the full range of discrete optimization theory (which includes graph algorithms, linear and integer programming, as well as the toolbox of combinatorial optimization with its geometric and numbertheoretic components), as well as the application areas from mathematical operations research, planning and scheduling, etc.

In the last fifteen years, Berlin has developed into an internationally visible and acclaimed place for graduate education in discrete mathematics and optimization. This is due not only to activities of individual faculty members and separate groups, but even more so to joint activities of the research groups in discrete mathematics and optimization at FU, HU, TU, and ZIB, including the RTGs Algorithmic Discrete Mathematics (1990–1998) and Combinatorics, Geometry and Computation (2000–2005) and their weekly colloquium, block courses, summer schools, as well as the regular Berlin Algorithms Day (BAT).

The following gives a brief sketch of main components of the Berlin research activity in discrete mathematics and optimization. In discrete mathematics, a well coordinated Berlin research agenda concerns the strengthening and broadening of the various (geometric, topological, probabilistic) tools and methods that are available in a discrete mathematics framework. This method orientation is also reflected in the RTG Methods for Discrete Structures.

For example, concerning discrete geometric methods the agenda spans and connects widely recognized work in polyhedral theory (Grötschel, Ziegler), discrete and algorithmic geometry (Alt, Felsner, Rote), and discrete differential geometry (Bobenko, Pinkall, Polthier, Sullivan); the last topic may be seen as a Berlin speciality, which is also being developed within a DFG Research Group Polyhedral Surfaces at TU Berlin. It amounts to a firm link between discrete mathematics and differential geometry.

In discrete optimization (Grötschel, Möhring, Skutella), a substantial effort is being put into the development of mathematical tools (such as scheduling algorithms and integer programming techniques) to the point where their impact can be seen on genuine industrial applications. A large part of the Berlin effort in this direction is based at the ZIB department of combinatorial optimization and the Application Area B Traffic and Communication Networks of the DFG Research Center MATHEON.