Commutative algebra is a branch of Mathematics that studies commutative rings and their ideals and modules. It has its roots in the end of 19th century when algebraic methods were developed to study structures and problems coming from number theory and invariant theory.

One of the main objects of interest in commutative algebra is the ring of polynomials (in one or more variables) over a field. Ideals in this ring naturally correspond to algebraic varieties, so commutative algebra is also a natural counterpart to algebraic geometry. Varieties such as curves and surfaces can be studied and sometimes better understood by looking at the associated algebraic structures, for example the ring of coordinates or the local rings at the points.

Commutative algebra is living a tantalizing moment. Recently, many long-standing conjectures have been proved or disproved. For example, the Eisenbud-Goto Conjecture which relates three important invariants of a homogeneous prime ideal (degree, dimension, and Castelnuovo-Mumford regularity) was shown to be false. On the opposite side, Stillman’s conjecture has been settled in the affirmative providing uniform bounds (independent on the number of variables) on the projective dimension of an ideal.

Finally, commutative algebra has found new interesting applications in Cryptography. New classes of cryptographic primitives have been constructed by using multivariate polynomial systems over finite fields. These schemes are particularly interesting nowadays because they are supposed to be resistant to attacks performed with a quantum computer.

The algebra group in Genoa is the largest and strongest group in commutative algebra in Europe and one of the strongest in the world. It features several researchers who are actively involved in the previously mentioned research topics. The members of the group are regularly invited to international conferences and workshops as speakers, and their results have been published in prestigious journals.