This research area comprises algebraic geometry, arithmetic geometry, and number theory. For many decades, all the three mentioned fields occupy a distinguished position at the very heart of mathematics. Moreover, the mutual interaction between the three fields has strongly stimulated the research in this area. Driving forces in the research of algebraic geometry are the minimal model program for higher dimensional algebraic varieties, as well as breakthroughs in moduli theory. In arithmetic geometry, arithmetic intersection theory culminating in arithmetic Riemann-Roch-type theorems and its applications to diophantine problems are at the forefront of present research. In algebraic number theory, the research is primarily fostered by the p-adic Langlands program, whereas analytic number theory has been fundamentally influenced by the groundbreaking results on measure rigidity in ergodic theory.

The respective research groups at the three Berlin universities cover a wide range of current research topics in this area. At FU, research focuses on the combining of algebraic and discrete/combinatorial geometry (Altmann), on the study of moduli spaces of principal bundles, in particular the moduli stack of shtukas (Schmitt), and on the classification of certain singular threefolds (Jahnke). At HU, research focuses on the investigation of birational and enumerative properties of parameter spaces in algebraic geometry, with an emphasis on the moduli space of curves (Farkas), on the understanding of the relevant categories of representations in the framework of the p-adic Langlands program (Große-Klönne), on extensions/applications of Arakelov geometry as well as fundamental conjectures in the analytic number theory of modular forms (Kramer), and on the construction of examples of abelian varieties over number fields with prescribed Tate-Shafarevich groups or with Mordell-Weil groups of large rank (Kloosterman). At TU, the chair in succession of Pohst is in the process of being refilled. The position has been announced in the area of algebra and/or number theory with possible collaborations with other fields represented at TU (such as algorithmic discrete mathematics or computer science).

Several researchers of this area participate in the SFB 647 Space, Time, Matter and the SPP 1489 Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory. All the scientists have been engaged in the RTG 870 Arithmetic and Geometry, which ended in December 2010.