  Let $X$ be a $d\times d$ symmetric random matrix with independent but
non-identically distributed Gaussian entries. It has been conjectured by
Lata\l{a} that the spectral norm of $X$ is always of the same order as the
largest Euclidean norm of its rows. A positive resolution of this conjecture
would provide a sharp understanding of the probabilistic mechanisms that
control the spectral norm of inhomogeneous Gaussian random matrices. This paper
establishes the conjecture up to a dimensional factor of order $\sqrt{\log\log
d}$. Moreover, dimension-free bounds are developed that are optimal to leading
order and that establish the conjecture in special cases. The proofs of these
results shed significant light on the geometry of the underlying Gaussian
processes.
