What are the implications for the AdS/Cft program if AdS is unstable? To my understanding recent progress in the study of the non linear stability of AdS spacetime suggest that $AdS$ might be unstable.
If this is true, what are the physical and mathematical implications for the $AdS/Cft$ approach?
The stability of an spacetime warranties that that sufficiently small perturbations remain small. In the case of Minkowski the stability theorems proved by Christodoulou and Klainerman show that sufficiently small perturbations not only remain small but decay to zero with time in any compact region (this stronger type ofstability is called asymptotic stability). 
In the case of $AdS$ numerical and certain results in nonlinear partial differential equations suggest the posibility that the following conjecture might be true:
The $AdS_{d+1}$ space (for d ≥ 3) is unstable against the
formation of a black hole for a large class of arbitrarily small perturbations.
Now, the $AdS/Cft$ correspondence is a conjectured relationship between two kinds of physical theories. On one side of the correspondence are conformal field theories (CFT) which are quantum field theories, including theories similar to the Yang–Mills theories that describe elementary particles. On the other side are Anti-de Sitter spaces (AdS) which are used in theories of quantum gravity, formulated in terms of string theory or M-theory.
In this program there are some examples that one can relate the formation of black holes in the bulk to certain thermodynamic properties of the conformal field theory. In particular,the presence of black holes can be seen as some thermalization of the field theory.
How would the results on stability be interpreted in the dual conformal picture?
Is there some meaningful thermodynamical process that explains the instability at the classical level?
 A: The question of AdS (in)stability is indeed a hot topic in current research of the AdS/CFT correspondence. It is a field that ties together many interesting subjects: Gravity in AdS (i.e a confining box), thermalization in QFTs, the theory of non-linear differential equations and their perturbative treatment, turbulence etc. This explains the explosion of works in this direction in recent years.
The topic arose from a seminal paper by Bizon & Rostoworowski in 2011, where they presented numerical evidence that the pure AdS solution of a gravity plus scalar system is unstable towards black hole formation for a certain class of initial conditions, namely certain Gaussian distributions. Instability means that even when one makes this initial perturbation arbitrarily small, a black hole will ultimately form. So there is no lower bound on the size of the perturbation which leads to black hole formation. They interpreted this as evidence towards a more general instability of AdS. Similar results where subsequently found for the AdS solutions of other systems: pure gravity, complex scalar plus gravity, Einstein-Maxwell. However, today the picture is more refined and it looks like that AdS has large islands of stability in the space of initial configurations.
Firstly, both from the field theory and from the gravity perspective, instability of AdS is not very surprising.
Via the AdS/CFT correspondence the question of AdS (in)stability is linked to the question of equilibration and thermalization. More precisely, which initial states in the dual field theory thermalize (on a certain timescale)? Is it therefore surprising that the question of AdS (in)stability is complex? Not at all, in quantum field theory the question of thermalization is extremely poorly understood, there is not even a proper definition of what thermalization is. How do you measure how close a density matrix is close to a thermal one? Which observables have to look thermal in order to say that a system has thermalized. Is there some kind of coarse-graining necessary, i.e. a partial trace over some subset of the Hilbert space that gets rid of the non-thermal information (that cannot be lost in a unitary time evolution). All these unclarities illustrate that studying thermalization from a bulk perspective should be non-trivial.
From a bulk perspective, the following intuition clarifies why instability of AdS is not at all surprising: Due to the boundary and the attractive effective gravitational potential there is no dissipation by dispersion (unlike the Minkowski and de Sitter cases). The boundary acts a a mirror. If one ads a finite excitation to a system in this box the system is expected to explore all configurations consistent with the conserved quantities and eventually the excitation will find itself within its own Schwarzschild radius and collapse.
Interestingly, Bizon and Rostoworowski found that instability in their model was due to the growth of so-called secular terms, resonances in the spectrum whose amplitude grows with time leading to turbulent transfer of energy to higher momentum modes and therefore to smaller scales which eventually leads to collapse.
In more recent studies, stable solutions have been found: e.g. Maliborski & Rostoworowski (2013); Buchel, Lehner & Liebling (2013). They represent field theory states that are small perturbations from the vacuum that do not thermalize. The interpretation is however difficult as the problem of thermalization is not well-understood from a field theory perspective, as mentioned above.
More recently, Craps, Evnin & Vanhoof have found an analytical approach to the subject (here and here) which closely resembles the renomalization group approach in perturbative QFT in order to get control over the secular terms arising in the time evolution which usually invalidate a perturbative treatment on short time-scales. This is highly welcome, as the numerical simulations are highly non-trivial and there are contradictory results from different groups which have not yet been resolved (e.g. these results versus these.) However, the analytical approach is still in development, so I do not know any concrete results yet. But I expect interesting results in this direction in the near future that might finally give us a better handle on thermalization of quantum systems.
