Core Courses
The three Core Courses provide a rigorous introduction to the most important objects and concepts of modern algebraic geometry and arithmetic geometry. The first semester focuses mainly on deepening knowledge in algebra, namely in commutative algebra, which is the basic prerequisite for algebraic geometry. The second and third semester then provides an introduction to the concepts and methods of modern algebraic geometry.
Prerequisites: group theory, rings and modules, field extensions and Galois theory.
Commutative Algebra
Rings, ideals, and modules
Prime, maximal, and primary ideals
Primary decompositions
Noetherian and Artinian rings
Discrete valuation rings and Dedekind domains
Hilbert‘s Nullstellensatz
Affine and projective varieties
Sheaves and their cohomology
Properties of morphisms of algebraic varieties
Fundamental examples of algebraic varieties
Hilbert functions and resolutions
Algebraic curves
Algebraic Geometry II
Schemes and their attributes
Line bundles and divisors
Cohomology of schemes
Complex manifolds and Hodge theory
Aspects of arithmetic geometry and of intersection theory
Spectrum of 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.
