“High Tc superconductivity” is referring to a modern mainstream in quantum matter physics that goes way beyond just superconductivity happening at high temperatures. This development started with the discovery of unreasonably sturdy superconductivity in copperoxides in the late 1980’s but has since then developed in one of the great mystery stories of modern physics. In short, it is about special solids where “chemical” conditions impose a physics on the electrons which is more like a (near) traffic jam than the extremely gaseous behaviors associated with the electron gas and BCS superconductivity of simple metals. These electron systems show new forms of highly quantum mechanical collective phenomena which appear to be quite ubiquitous. These are found in cuprates, organics, heavy fermion intermetallics and in the recently discovered iron superconductors, but they are theoretically poorly understood: for a recent perspective see the “consensus paper” written on request by the Nature editorial staff.
The Leiden group has a long history in this field, starting in the 1980’s adressing the “chemistry” in terms of the famous “ZSA” electronic structure , later quantified by the construction of LDA+U by Zaanen [34, 58]. The group pioneered the idea of uncoventional order in doped Mott insulators, predicting the so-called electronic stripes  which turned out later to be quite ubiquitous. The group spend much energy in the 1990’s trying to get a handle on the idea of “dynamical stripes”, quantum liquids which show strong stripe correlations at short distances : the origin of the name “stripe club”. The empasis of the theoretical research in this direction has since then turned towards the theory of quantum liquid crystals, but there is continuing interest in the “competing order” theme especially in relation to experiment. The group introduced the idea in 2009 that in pnictides the orbital ordering tendency could trigger quantum nematic order, an idea that has become quite popular . Another recent highight is the surprisingly rich structure of the topological defects in the electron stripes ordering seen by scanning tunneling specroscopy on copper-oxide surafecs [175, plaatje er uit halen !?].
The greatest mystery of all are however the “strange metals” that are found in all these systems, usually going hand in hand with the best superconductivity. These exhibit very simple physical properties like a linear resistivity which are however very hard to understand on basis of established theoretical wisdoms. The group has been pioneering the notion that this is governed by a new form of emergent “fermionic” scale invariance of the quantum many particle physics . This involves phenomenological work in close touch with experiment [105,155] while Zaanen is responsible for the designation “Planckian dissipation” for the anomalous dissipative properties of finite temperature quantum critical liquids . Another, untested notion in this context is the “quantum critical BCS’ scaling theory, where it is demonstrated that the BCS type superconductivity can lead to much higher Tc’s in a quantum critical metal as compared to a Fermi liquid . Such a behavior appears to be realized in the Hubbard model according to very modern “cluster DMFT” computations .
The bottom line is that through the thirty years of its history “high Tc” has acquired the status of central mystery in condensed matter physics, similar to the dark sector of cosmology or the nature of life in biology. Mankind is blind without equations in this realm of reality and given the regularity seen in the experiments it seems obvious that a new kind of mathematics is at work. In recent years it has become obvious from various developments that this mystery is somehow rooted in the dense many body entanglement at work in these electron systems. To tackle this exponential complexity (NP-hardness) a quantum computer is needed in principle. Inspired by the “quantum supremacy” of the latter we like to call the stuff found in the cuprates quantum supreme matter.
Despite this quantum supremacy such matter should still behave in a reasonable way, although new general principles are expected. The progress in recent years has much helped by the arrival of mathematical machinery that has much to say about these new principles. On the one hand, the quantum information inspired “big numerical machines” such as the tensor network algorithms have started to deliver. As an example, four leg Hubbard ladders can now be precisely computed using the DMRG tensor networks, showing that extremely large bond dimensions are required indicating very dense many body entanglement (see https://arxiv.org/abs/1907.11728. ) Turn this into the number-link.
The other game changer is the AdS/CFT or holographic duality [cite the book]. This mathematical machine was discovered in string theory as a computational device supposedly shedding light on quantum gravity. In a highly serendipitous development it turned into a machinery describing quantum systems littered with fermion signs that are in a precise sense describing the limit of maximal many body entanglement, absurdly by mathematically enumerating the properties of black holes. Although the jury is still out whether this limit is of a literal relevancy to high Tc and related phenomena it has been at the least a source of highly unusual questions to experiment. These are described under “Holography in the laboratory”. This agenda is increasingly inspiring the experimentalists and the case is rapidly developing that holography has put mankind somehow on the right track. In the Netherlands this has substantiated in a 2.5 MEuro block grant for an experiment dominated strange metal research program http://strangemetals.nl inspired on the holographic questions.
For instance, it is now within reach of experiment https://arxiv.org/abs/1807.10951 to prove or disprove that the famous linear resistivity as rooted in the “Planckian dissipation”  is an expression of the minimal viscosity first seen in the QCD plasma. Astonishing photoemission evidence [Sudi Chen’s paper, accepted by Science, also science 362, 62 (2018)] appeared for the strange metal to be a genuine quantum critical phase of matter. The case is developing that even along the nodal directions the “peaks” as seen in ARPES are not at all quasiparticles in the usual sense [van Heumen, unpublished] but instead more
closely related to the “holographic fermions” . Holography suggests an unreasonably rapid thermalization in densely entangled matter when it is highly excited https://arxiv.org/abs/1804.04735. This appears to be at work in the heavy ion collisions and first attempts are under way trying to observe this by pump-probe spectroscopy in the cuprates. Last but not least, holography renders an astonishingly natural explanation for the “intertwined order” in underdoped cuprates Cai et al, PRL 119, 181601. Specifically, including the background lattice one finds back the essence of the spin stripes [Andrade, Nature Phys. 14, 1049 (2018)] with the prediction that the mysterious “quantum critical sector” gets isolated. Holography is insisting that this is characterized by emerging charge conjugation symmetry. The prediction is that the Hall effect should vanish everywhere in such a holographic stripe phase and this appears to be confirmed in a very recent experiment https://arxiv.org/abs/1909.02491