  For the one-dimensional Schr\"odinger equation, we obtain sharp
maximal-in-time and maximal-in-space estimates for systems of orthonormal
initial data. The maximal-in-time estimates generalize a classical result of
Kenig--Ponce--Vega and allow us obtain pointwise convergence results associated
with systems of infinitely many fermions. The maximal-in-space estimates
simultaneously address an endpoint problem raised by Frank--Sabin in their work
on Strichartz estimates for orthonormal systems of data, and provide a path
toward proving our maximal-in-time estimates.
