We consider wave maps from the (1+d)-dimensional Minkowski space
into the d-sphere. Early numerical works by P. Bizo´n et al. on this model indi-
cate that in the supercritical case d ? 3, generic large data lead to finite time
blowup via certain explicitly known self-similar solution. In attempts to rigor-
ously confirm these observations, many results have been obtained over the past
two decades, beginning with the pioneering work of P. Aichelburg, R. Donninger
and B. Sch¨orkhuber. However, despite their groundbreaking nature, these works
establish stability results that are only local-in-space.
In this talk, we present a novel framework for studying spatially global stability
of self-similar solutions to semilinear wave equations. We apply this approach
to the wave maps setting described above, obtaining the first stability result that
is global-in-space, and thereby verifying the perturbative version of a conjecture
posed by P. Bizo´n and P. Biernat in 2015. At the and, we discuss further open
problems as well as the new mathematical challenges that our approach generates.

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