# Algebraic Geometry and Commutative Algebra

**Algebraic geometry** is the study of solutions of systems of polynomial equations with geometric methods. It provides a prime example of the interaction between algebra and geometry. Projective varieties are covered by affine varieties, which correspond to polynomial algebras over a field. To an arbitrary commutative ring, Grothendieck associated an affine scheme; gluing these, one obtains schemes (and recovers varieties). Nowadays, stacks and algebraic spaces play a fundamental role in the study of moduli spaces.

**Commutative algebra** provides the foundation for algebraic geometry, but can of course also be studied in its own right. Some of the topics of current interest at KTH are Gorenstein and level algebras and Betti diagrams of Cohen–Macaulay modules.

We run a weekly Algebra & Geometry seminar (also including algebraic topology) and sometimes also more focused seminars.