Abstract
Over the last twenty years, simulations of galaxy formation from cosmological initial conditions have become an invaluable resource for advancing modern galaxy formation research. These simulations have not only successfully reproduced stellar masses and the diverse morphological types of galaxies observed at redshift 0 but have also captured a wide array of other properties. While these models serve as a crucial link between cosmology and the observed galaxy population, they still face significant limitations – particularly at small scales. The modeling of feedback effects from stars and active galactic nuclei (AGN) remains highly oversimplified, largely dictated by resolution constraints. As a result, the simulated gas flows often differ substantially from those produced in coarse-grained, high-resolution models. In this presentation, I will highlight recent progress made to address these challenges. First, I will examine the impact of AGN feedback in the most massive halos and demonstrate how simulations are instrumental in interpreting the latest XRISM observations of cool-core galaxy clusters. Next, I will explore AGN feedback in lower-mass systems, emphasizing the role of multi-phase gas with insights gained from recent JWST observations of galactic outflows, and discuss our modeling approach. Finally, I will introduce a multi-fluid formulation for multi-phase gas and illustrate how this method can help overcome the limitations of current simulations, ultimately reconciling small-scale and large-scale models.

Abstract
We explore the formation of intermediate mass black holes (IMBHs), potential seeds for supermassive black holes (SMBHs), via runaway stellar collisions for a wide range of star cluster densities and metallicities. Our sample of isolated (>1400) and hierarchical (30) simulations of massive JWST z~10 analogue star clusters with up to 1.8 million stars includes collisional stellar dynamics, stellar evolution, and post-Newtonian equations of motion for black holes using the BIFROST code. High stellar wind rates suppress IMBH formation at high metallicities (above 0.2 of the solar value) and low collision rates prevent their formation at low densities. The assumptions about stellar wind loss rates strongly affect the final IMBH masses (6000 vs 25000 solar masses). We present fitting formulae for IMBH masses as a function of host star cluster properties, and formulate a simple model for the cosmic IMBH formation rate density. We highlight the current model uncertainties and emphasize the need for collaboration between the galaxy formation and stellar communities for future progress in IMBH science. Finally, we find that the characteristic gravitational wave (GW) fingerprint of hierarchical star cluster assembly is a close-to-equal mass GW merger in the IMBH mass range.