  The Matrix Spencer Conjecture asks whether given $n$ symmetric matrices in
$\mathbb{R}^{n \times n}$ with eigenvalues in $[-1,1]$ one can always find
signs so that their signed sum has singular values bounded by $O(\sqrt{n})$.
The standard approach in discrepancy requires proving that the convex body of
all good fractional signings is large enough. However, this question has
remained wide open due to the lack of tools to certify measure lower bounds for
rather small non-polyhedral convex sets.
  A seminal result by Batson, Spielman and Srivastava from 2008 shows that any
undirected graph admits a linear size spectral sparsifier. Again, one can
define a convex body of all good fractional signings. We can indeed prove that
this body is close to most of the Gaussian measure. This implies that a
discrepancy algorithm by the second author can be used to sample a linear size
sparsifer. In contrast to previous methods, we require only a logarithmic
number of sampling phases.
