  We consider $N$ non-interacting fermions in an isotropic $d$-dimensional
harmonic trap. We compute analytically the cumulative distribution of the
maximal radial distance of the fermions from the trap center at zero
temperature. While in $d=1$ the limiting distribution (in the large $N$ limit),
properly centered and scaled, converges to the squared Tracy-Widom distribution
of the Gaussian Unitary Ensemble in Random Matrix Theory, we show that for all
$d>1$, the limiting distribution converges to the Gumbel law. These limiting
forms turn out to be universal, i.e., independent of the details of the
trapping potential for a large class of isotropic trapping potentials. We also
study the position of the right-most fermion in a given direction in $d$
dimensions and, in the case of a harmonic trap, the maximum momentum, and show
that they obey similar Gumbel statistics. Finally, we generalize these results
to low but finite temperature.
