At FU, the research focuses on combinatorial algebraic geometry, in particular related to deformation theory (Altmann). At HU, there are groups working in complex geometry (Leiterer), algebraic geometry (Farkas), arithmetic algebraic geometry (Kramer), and in number theory related primarily to the Langlands program (Zink) and the theory of automorphic forms (Kramer). All these groups are actively integrated into the International RTG Arithmetic and Geometry, a joint initiative with ETH and the University of Zurich. At TU, research is concerned with computational number theory and computational algebraic geometry (Hess, Pohst).

The current research activities of the algebraic geometers comprise the presently very active research in mirror symmetry using derived categories and Fourier-Mukai transforms. This in turn is based on a deep knowledge of the theory of deformations and resolution of singularities, e.g., using toroidal methods. In addition, there is work contributing to the higher dimensional birational classification program. All these activities have strong links to mathematical physics. Thus there is close cooperation with the SFB 647 Space, Time, Matter. Moreover, the research in complex geometry is closely related to the work of the algebraic geometers and provides important analytical tools.

In arithmetic algebraic geometry fundamental work in Arakelov geometry has been developed. This has applications to the study of rational points (and higher dimensional cycles) on algebraic varieties over number fields, extending the techniques used by Faltings in his famous proof of the Mordell conjecture. In addition, problems from transcendence theory can be investigated with these new tools. Another important field of applications of this newly developed theory is the theory of automorphic forms and Shimura varieties. In particular, the research work in arithmetic algebraic geometry has strong links to global analysis and differential geometry.

In number theory the research work is two-fold: The research group working on basic problems in number theory is primarily concerned with questions arising from the Langlands program and the theory of modular/automorphic forms. Here, questions about the classification of automorphic representations and Galois representations and their interplay are of central interest, as well as questions on the arithmetic of modular and automorphic forms. The program to extend the Langlands correspondence to the p-adic setting is also part of the research. In particular, the latter work asks for new techniques which are of functional analytical flavor. On the other hand, the KANT group at TU develops efficient algorithms for number fields, algebraic curves and related objects with special focus on local and global class field theory, K-theory, complex multiplication, and diophantine equations. Applications to coding theory and cryptography are also considered. There are strong links to numerical analysis, computer science, and discrete mathematics.