The commutative algebra group in Genova has research interests that include homological algebra, algebraic combinatorics, algebraic geometry and representation theory. More specifically, some topics that are actively pursued by faculty members of this group are:
Gröbner deformations, generic initial ideals and all the related computational applications.
Homological algebra (Betti numbers, Castelnuovo-Mumford regularity, Koszul complex, local cohomology, etc.).
Singularities of local and graded rings (Gorenstein, Cohen-Macaulay, Buchsbaum, Koszul, etc.)
Singularities related to the Frobenius homomorphism, and methods in positive characteristic.
Determinantal rings, Grassmannians, ideals generated by Pfaffians etc.
Stanley-Reisner rings and the study of the singular homology of a simplicial complex.
The set-theoretic definition of algebraic varieties.