Commutative Algebra, the study of commutative rings and their modules, has a long and prestigious history which started to really take off with the works of David Hilbert in the late 19-th and early 20-th century and Emmy Noether in the 1920s. When Bruno Buchberger discovered the algorithm for computing Gröbner bases in 1965, an entirely new dimension opened up for this traditional part of mathematics, namely the possibility to symbolically calculate with these objects.

In this school we start by introducing the participants to the basics of the theory of Gröbner bases, Buchberger’s algorithm and Hilbert functions. Then we branch out and convey a wide range of applications that these fundamental techniques have produced in the last decades. These include applications to questions in commutative algebra itself, such as the effective manipulation of ideals and modules, elimination, and polynomial system solving.

A natural domain for applying these techniques is algebraic geometry. Here we bring the theory to life by looking at finite sets of points and the connection between their algebraic invariants (such as the Hilbert function) and their geometric properties.

Closely related is the application to coding theory. Linear codes are finite sets of points in affine spaces over a finite field. Once again the techniques of computational commutative algebra allow us to compute and study deeper invariants such as the minimal distance or the weight enumeration polynomials of a code.

A further application domain is post-quantum cryptography. In view of the rapid progress made in developing quantum computers, this area has a high degree of urgency. The task is to invent encryption methods which are expected to be secure against adversaries possessing powerful quantum computers. Several approaches to this task have been proposed and will be studied, and once again Gröbner basis methods have an important role to play.

The goal of the school is to first lay a solid foundation by teaching the theory of Gröbner bases and Hilbert functions and their computation. Then we introduce the participants to various application areas and provide them with a set of tools such that they can later devise applications to their own problems in various research areas, including, but not limited to, commutative algebra, algebraic geometry, coding theory, and post-quantum cryptography.