2. Algebra and number theory, algebraic and arithmetic geometry.
The two basic courses provide a rigorous introduction to the most important objects and concepts of modern algebraic geometry and number theory. The first semester focuses mainly on deepening the 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:

• Commutative algebra
◦ rings, ideals, and modules
◦ prime ideals, maximal ideals, and primary ideals
◦ primary decompositions
◦ Noetherian and Artinian rings
◦ discrete valuation rings and Dedekind domains

• Algebraic geometry
◦ varieties
◦ schemes and sheaves
◦ cohomology theory
◦ curves and surfaces