Algebra and geometry

Algebraic geometry, the study of solutions of systems of polynomial equations, 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.

Algebraic topology provides another example of the interaction. Topology is a different, more flexible, kind of geometry, where objects are built out of discs and spheres rather than represented by equations. The search for algebraic invariants of such geometrical representations of spaces led to the field of algebraic topology. Efficient tools to study such invariants of topological spaces are provided by homotopy theory, whose techniques are nowadays used to understand many other types of mathematical objects.

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

 All three subjects are represented at KTH:

ALGEBRAIC GEOMETRY

   * Sandra Di Rocco
   * Carel Faber
   * Katharina Heinrich
   * Jack Hall
   * Dan Laksov
   * Anders Lundman
   * Dan Petersen
   * David Rydh
   * Gustav Sædén Ståhl
   * Roy Skjelnes

ALGEBRAIC TOPOLOGY

   * Tilman Bauer
   * Wojciech Chachólski
   * Sebastian Öberg

COMMUTATIVE ALGEBRA

   * Mats Boij
   * Ornella Greco