Stillman's conjecture (now a theorem by Ananyan-Hochster) shows that for all d_1,...,d_r there exists D such that any ideal generated by polynomials f_1,...,f_r of degrees d_1,...,d_r has projective dimension at most D, no matter the polynomials or the number of variables these polynomials are defined in. By Erman-Sam-Snowden, similar uniformity problems can be solved provided that we know the space of r-tuples of homogeneous polynomials in degrees d_1,...,d_r is Noetherian up to the action of the general linear group.

In this talk, we will discuss some of the above, and prove Noetherianity of the space of cubic polynomials. This gives the first known finiteness result in degree 3 or above. We prove Noetherianity by means of the concept of q-rank of a cubic polynomial, which is strongly related to the concept of strength used by Ananyan-Hochster in their proof of Stillman's conjecture. Speaker(s): Rob Eggermont (University of Michigan)