It is by now well established that, upon cooling, local dynamical fluctuations in supercooled fluids grow ever more intense. This has ignited an important line of work attempting to understand the behaviour of supercooled liquids through the lens of critical phenomena. Indeed, many studies have demonstrated that dynamical susceptibilities, and an associated dynamic correlation length, grow upon cooling, in close analogy with the divergence of the correlation length on approach to a standard critical point.
Yet, in this talk, I will argue that the spatial extent of cooperative motion does not grow monotonically, and that this non-monotonicity is itself the hallmark of the avoided mode-coupling transition in finite dimensions. The conventional understanding states that a dynamical correlation length, measured at the $\alpha$-timescale, increases monotonically as supercooled liquids approach the mode-coupling crossover $T_c$ (and below), in line with the true divergence predicted by mean-field theory. It is commonly believed that this divergence is avoided in finite dimensions because of non-perturbative processes, namely activated, hopping-like events that lie outside the scope of mode-coupling theory altogether. I will argue that this is not the case: the transition is avoided already by critical fluctuations, perturbatively built on top of the mean-field solution. These fluctuations, however, are not concentrated on the $\alpha$-timescale but on the $\beta$-timescale, and it is in this dynamical window, not the $\alpha$-window, that the avoidance of the transition must be sought.
Turning to a model glass-forming system, I will show that the correlation length extracted from the $\beta$-regime behaves in exactly the way the local fluctuations do not: it rises on cooling, peaks near $T_c$, and then \emph{decreases} on further cooling, even as local dynamical heterogeneity keeps intensifying. The resolution of the paradox lies in a crossover in the character of cooperative motion itself: from spatially extended, maximally collective relaxation at $T_c$, to progressively more compact, localized relaxation events below it.
This non-monotonic length is quantitatively reproduced by stochastic beta-relaxation theory, the extension of mode-coupling theory to finite dimensions, which predicts precisely such an avoided dynamical transition. Our results are, to our knowledge, the first direct spatial evidence for the avoided-transition scenario at the mode-coupling crossover. They further establish the peak of the $\beta$-regime correlation length as a sharp, robust and long-sought-after definition of $T_c$ itself.