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, p-adic 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 p-adic Langlands program, whereas analytic number theory has been fundamentally influenced by the groundbreaking results on measure rigidity in ergodic theory.

Berlin research groups

The research groups in algebraic and arithmetic geometry in Berlin are among the most visible in theoretic field in Europe and very attractive for students from all over the world. Currently our activity is supported by the newly established RTG 2965 "From Geometry to numbers: Moduli, Hodge theory, rational points" (between the HU Berlin and Hannover), as well as by the two ERC Advanced Grants of Gavril Farkas (Syzygy) and Bruno Klingler (Tame Geometry). Research in the area at the three universities cover a wide range of topics lying at the heart of algebraic and arithmetic geometry. At HU, research focuses on the investigation of geometric properties of parameter spaces in algebraic geometry, with an emphasis on the moduli space of curves, on the study of syzygies of algebraic varieties with applications to geometric group theory (Farkas), on the study of singularities and moduli spaces via perverse sheaves, D-modules and representation theory (Krämer), on abelian varieties, vector bundles and their moduli spaces (Ortega), on the study of periods of families of algebraic varieties, their transcendence properties and their Hodge theoretic incarnation (Klingler), on the understanding of the relevant categories of representations in the framework of the p-adic Langlands program (Große-Klönne). At FU, research focuses on the combining of algebraic and discrete/combinatorial geometry and on the study of moduli spaces of principal bundles and invariant theory in algebraic geometry (Schmitt).

Gaetan Borot	Mathematical Physics Mathematical physics can be approached in many different ways. Here, it is a cross-section of traditional areas of mathematics, involving asymptotic analysis, low-dimensional geometry and geometry of moduli spaces, algebraic structures and combinatorics, which are often (at least historically) enlightened by ideas born in contemporary physics, in particular statistical physics, quantum field theory and string theory. More specific areas of active research are the enumerative geometry of complex curves, the related (quantum) geometry of moduli spaces of geometric structures (relation to RTA 2); (supersymmetric) gauge theories and conformal field theories; random matrix theory (relation to RTA 3) and free probability; the combinatorics and geometry of flat or hyperbolic surfaces (relation to RTA 4 and 5); and the deep relations and unifying perspectives behind these problems.	HU Berlin
Alexandru Constantinescu	Commutative Algebra and Algebraic Geometry Constantinescu’s fields of interest are algebraic geometry, commutative algebra, and discrete mathematics, with a particular focus on how these three areas connect and interact. He is especially interested in Hilbert schemes, Castelnuovo-Mumford regularity, the theoretical aspects of Gröbner degenerations, the interaction between deformation theory and discrete structures (such as polyhedra, discrete semigroups, and simplicial complexes), and the connections between commutative algebra and geometric group theory.	FU Berlin
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 one-dimensional 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 Brill-Noether type problems. Connections of algebraic geometry to neighboring fields like geometric group theory or topology are also studied.	HU Berlin
Elmar Große-Klönne	Local Fields and Representation Theory In modern Number Theory and Arithmetic Geometry, studying objects defined over local fields (like the field of p-adic 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 p-adic 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 Geometry There 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 Newton-Okounkov 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
Bruno Klingler	Arithmetic Geometry and Hodge Theory Hodge theory (complex or p-adic) 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 p-adic 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
Thomas Krämer	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
Angela Ortega	Abelian and Prym varieties and their moduli. Moduli spaces of vector bundles over curves Ortega's research interests are, on one side the geometry of the moduli space of vector bundles on algebraic curves and Brill-Noether 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 Torelli-type Theorems for Prym varieties.	HU Berlin
Alexander Schmitt	Moduli Spaces in Algebraic Geometry Algebraic 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 p-adic Hodge Theory, D-modules 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