This is the second of two talks on joint work of Tigran Ananyan and the speaker. The main result is that there is a positive integer B(n,d) such that given n forms F_i of positive degree at most d in a polynomial ring R in N variables over an algebraically closed field K of characteristic 0 or greater than d (note that B = B(n,d) does NOT depend on N), there exists a regular sequence of forms g_1, ..., g_h, where h is at most B, of degree at most d such that the F_i are in the ring K[g_1, ..., g_h]. This places the F_i in a "small" polynomial subring of R over which R is faithfully flat (in fact, free). This implies Stillman's conjecture that the projective dimension of the ideal I generated by the F_i is bounded independent of N in the case where the characteristic is 0 or large: it will be at most B(n,d). The authors had earlier shown this only up to degree 4. One can also bound all numerical data about the primary decomposition of I independent of N. The authors have also obtained these results for d = 2 and d = 3 with no restriction on the characteristic, and in degree 4 except in characteristic 2. Many open questions remain. Speaker(s): Mel Hochster (University of Michigan)