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This article is devoted to the energy critical hyperbolic Yang--Mills systemin the $(4+1)$ dimensional Minkowski space, which is considered by the authorsin a sequence of four papers. The final outcome of these papers is twofold: (i)the Threshold Theorem, which asserts that global well-posedness and scatteringhold for all topologically trivial initial data with energy below twice theground state energy, and (ii) the Dichotomy Theorem, which for larger data inarbitrary topological classes provides a choice of two outcomes, either aglobal, scattering solution or a soliton bubbling off. In the last case, thebubbling off phenomena can happen either (a) in finite time, triggering afinite time blow-up, or (b) in infinite time. Our goal here is to describethese results, and to provide an overview of the flow of ideas within theirproofs.

The Threshold Theorem for the $(4+1)$-dimensional Yang--Mills equation: an overview of the proof