Explore a detailed physics seminar from Harvard CMSA where Prahar Mitra from the University of Cambridge presents groundbreaking research on new static solutions in gauged supergravity and their implications for N=4 Super Yang-Mills theory. Delve into the construction of thermal phases in N=4 SYM at finite chemical potential through the lens of AdS/CFT correspondence, examining how these solutions dominate the micro-canonical ensemble and contribute to understanding AdS black hole entropy. Learn about the distinct truncations of gauged supergravity, their uplift to type IIB supergravity solutions, and gain insights into the phase space of SYM thermal states with three arbitrary chemical potentials. Discover the current understanding of hairy supersymmetric black hole solutions in this theoretical framework, drawing from research published in arXiv:2207.07134.