Effective bounds for prime ideals

This talk is based on a joint work with Yihui Liang and Cheng Meng. I will present explicit bounds for the projective dimension and the Castelnuovo-Mumford regularity of homogeneous ideals in the setting of Stillman's Conjecture and its solution by Ananyan and Hochster. As a corollary, in the spirit of the Eisenbud-Goto conjecture and its counterexamples, one derives explicit bounds for the Castelnuovo-Mumford regularity of homogeneous prime ideals solely in terms of their multiplicity.