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. 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, breakthroughs in moduli theory, and Hodge theory. At the borderline between algebraic and arithmetic geometry are the theory of motives. In arithmetic geometry, arithmetic intersection theory culminating in arithmetic Riemann-Roch-type theorems and its applications to 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 respective research groups at the three Berlin universities cover a wide range of current research topics in this area. 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 (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), and on new foundational challenges in Arakelov geometry as well as fundamental problems in the theory of modular and automorphic forms, in particular related to questions in analytic number theory (Kramer). At FU, research focuses on the combining of algebraic and discrete/combinatorial geometry (Altmann), on the study of moduli spaces of principal bundles and invariant theory in algebraic geometry (Schmitt).
The three Basic Courses provide a rigorous introduction to the most important objects and concepts of modern algebraic geometry, arithmetic geometry and number theory. The first semester focuses mainly on deepening knowledge in algebra, namely in commutative algebra, which is the basic prerequisite for algebraic geometry and number theory. The second semester then provides an introduction to the concepts of modern algebraic geometry.
Prerequisites: group theory, rings and modules, field extensions and Galois theory.
Outline of the contents:
◦ Rings, ideals, and modules
◦ Prime, maximal, and primary ideals
◦ Primary decompositions
◦ Noetherian and Artinian rings
◦ Discrete valuation rings and Dedekind domains
◦ Affine and projective varieties
◦ Sheaves and schemes
◦ Line bundles and divisors
◦ Cohomology of sheaves
◦ Local and global fields
◦ Aspects of arithmetic geometry
Spectrum of Advanced Courses
The advanced courses regularly offered include Abelian Varieties, Algebraic Stacks, Arakelov Geometry, Birational Geometry of Algebraic Varieties, Class Field Theory, Complex and p-adic Hodge Theory, D-modules in Algebraic Geometry, Elliptic Curves and Cryptography, Moduli Spaces, Shimura Varieties, Syzygies of Algebraic Varieties, Teichmüller Theory, Theory of Automorphic Forms.