Algebraic Geometry, Arithmetic Geometry and Number Theory
This research area comprises algebraic geometry, arithmetic geometry, and number theory. For many decades, all the three mentioned fields have occupied a distinguished position at the very heart of mathematics. Progress in the field has been repeatedly recognized with the most important prizes in mathematics. Driving forces in the research of algebraic geometry are the minimal model program for higher dimensional algebraic varieties, breakthroughs in moduli theory, Hodge theory and tropical geometry. At the borderline between algebraic and arithmetic geometry lies the theory of motives. In arithmetic geometry, padic geometry, the study of Shimura varieties and diophantine problems are at the core of present research, along with the study of rational points over various fields, and of properties of fundamental groups. In algebraic number theory, the research is primarily fostered by the padic Langlands program, whereas analytic number theory has been fundamentally influenced by the groundbreaking results on measure rigidity in ergodic theory.
Berlin research groups
Gavril Farkas  Algebraic Geometry A significant part of Farkas' research concerns the geometry and topology of the fundamental moduli spaces that appear in algebraic geometry, namely the moduli space of algebraic curves and that of abelian varieties. Algebraic curves are among the most uniquitous objects in mathematics, being studied with different methods in algebraic geometry (as onedimensional algebraic varieties), complex geometry (as Riemann surfaces) and dynamics. Further important aspects of research concerns the geometric study of the equations and syzygies of algebraic varieties, enumerative geometry, K3 surfaces and BrillNoether type problems. Connections of algebraic geometry to neighboring fields like geometric group theory or topology are also studied. 
HU Berlin 
Elmar GroßeKlönne  Local Fields and Representation TheoryIn modern Number Theory and Arithmetic Geometry, studying objects defined over local fields (like the field of padic numbers) plays a decisive role. Of particular prominence is the speculative local Langlands program which tentatively describes representations of Galois groups of local fields in terms of representations of padic reductive groups. Various Hecke Algebras show up; Geometry, Representation theory and Combinatorics merge in an exciting way. 
HU Berlin 
Christian Haase  Discrete Methods in Algebraic GeometryThere is a rich interplay of algebraic geometry, polyhedral geometry and combinatorics. Deep theorems from algebraic geometry and commutative algebra are applied to combinatorial problems. Conversely, algebraic problems are reduced to considerations in discrete geometry. For instance, the theory of toric varieties is an established and very active field of research. Methods from tropical geometry and NewtonOkounkov theory have widened the scope of these methods. Applications come from matroids, auction theory, integer linear optimization and even the study of the expressivity of neural networks. 
FU Berlin 
Arithmetic Geometry and Hodge TheoryHodge theory (complex or padic) is one of the main tools for analyzing the geometry and arithmetic of algebraic varieties (that is, solution sets of algebraic equations) over the complex or the padic numbers. It occupies a central position in mathematics through its relations to differential geometry, algebraic geometry, differential equations and number theory. It is currently undergoing many exciting developments, in relation to tame geometry and functional transcendence. 
HU Berlin 

Algebraic Geometry The basic objects of algebraic geometry are algebraic varieties, the sets of solutions to systems of polynomial equations in several variables. Depending on the context these can be studied with tools from many different areas, leading to rich interactions between geometry, topology, algebra and number theory. Important research topics in Berlin include algebraic curves, abelian varieties and their moduli spaces, Hodge theory and rational points on varieties over number fields, and the study of singularities via perverse sheaves and representation theory. 
HU Berlin 

Abelian and Prym varieties and their moduli. Moduli spaces of vector bundles over curvesOrtega's research interests are, on one side the geometry of the moduli space of vector bundles on algebraic curves and BrillNoether theory, and on the other side, Abelian and Prym varieties and their moduli. More precisely, she works on problems related to the structure of the Prym map and Torellitype Theorems for Prym varieties. 
HU Berlin 

Moduli Spaces in Algebraic GeometryAlgebraic geometry studies systems of polynomial equations and their sets of solutions in affine or projective space (these solution sets are called algebraic varieties). If the equations are linear, the solution set is an affine space, and the only interesting invariant is its dimension. For equations of higher degree, one quickly realizes that explicit manipulations of the equations do not lead very far, so that one needs a much more abstract machinery for classifying algebraic varieties. It often turns out that the isomorphism classes of certain algebraic varieties can be identified in a natural way with the points of another algebraic variety. The latter is then called a moduli space. For example, it is classically known that two given ordered sets of four distinct points on the projective line may be transformed into each other by a linear coordinate change if and only if their cross ratios are equal. So, the corresponding moduli space is the field minus 0 and 1 or, equivalently, the projective line minus the points 0, 1, and infinity. Moduli spaces arise also from other problems in algebra, such as studying matrices under conjugation or more complicated systems of matrices, called quiver representations. Such objects have recently found interesting applications to Big Data and neural networks. 
FU Berlin 
Core Courses
The BMS Core Courses in Area 2 and details about their content can be found under "Course Program":
Area 2  Core Courses
Advanced Courses
The advanced courses regularly offered include Abelian Varieties, Algebraic Stacks, Birational Geometry of Algebraic Varieties, Class Field Theory, Complex and padic Hodge Theory, Dmodules in Algebraic Geometry, Derived categories in algebraic geometry, Elliptic Curves and Cryptography, Moduli Spaces, Perfectoid spaces, Shimura Varieties, Syzygies of Algebraic Varieties, Teichmüller Theory.
The current advanced courses are listed under "Course Program":
Area 2  Advanced Courses