  When mathematical/computational problems reach infinity, extending analysis
and/or numerical computation beyond it becomes a notorious challenge. We
suggest that, upon suitable singular transformations (that can in principle be
computationally detected on the fly) it becomes possible to go beyond infinity
to the other side, with the solution becoming again well behaved and the
computations continuing normally. In our lumped, Ordinary Differential Equation
(ODE) examples this infinity crossing can happen instantaneously; at the
spatially distributed, Partial Differential Equation (PDE) level the crossing
of infinity may even persist for finite time, necessitating the introduction of
conceptual (and computational) buffer zones in which an appropriate singular
transformation is continuously (locally) detected and performed. These
observations (and associated tools) could set the stage for a systematic
approach to bypassing infinity (and thus going beyond it) in a broader range of
evolution equations; they also hold the promise of meaningfully and seamlessly
performing the relevant computations. Along the path of our analysis, we
present a regularization process via complexification and explore its impact on
the dynamics; we also discuss a set of compactification transformations and
their intuitive implications.
