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Strongly correlated quantum phenomena in low dimensions are particularly challenging to understand. In one and two dimensions strong quantum fluctuations lead the fundamental constituents of matter to organize in novel, often unpredictable ways. Phenomena such as non-abelian statistics, spin-charge separation, quantum number fractionalization, and topological states of matter can arise. These new forms of matter cannot be studied with the perturbative approaches commonly used to understand the physics of weakly interacting systems.

Integrability provides a unique tool able to access the non-perturbative physics present in strongly correlated systems. An integrable system possesses non-trivial conservation laws beyond the standard ones of energy and momentum. Integrability is surprisingly ubiquitous in one-dimensional systems, ranging from quantum wires, to spin chains, to cold atomic gases. The additional conservation laws constrain the physics of the system making it possible to access, in certain circumstances, non-perturbative information.

The workshop will be driven by the questions surrounding so-called "weak breaking of integrability" that is displayed by a wide spectrum of systems that are not actually integrable, but very nearly so. In particular: Does the weak breaking obliterate all of the features of integrability? Or is there a smooth crossover between integrable systems and fully non-integrable systems? These and similar questions are important from the theoretical standpoint, for instance to make contact with KAM theory, for technical reasons in order to go beyond Keldysh perturbation theory and develop theoretical tools to study non-equilibrium phenomena, but also most importantly because of the bearing on questions of thermalization in closed quantum systems. This nexus has come to the forefront of research because of recent key experiments in cold atomic gases.

Thus the goal of this workshop is to bring together members from the three different communities: Mathematicians working on integrability, theoretical physicists interested in the applications of integrability in low-dimensional quantum systems, in and out of equilibrium, and finally, experimentalists studying integrable and quasi-integrable systems. Bringing together the disparate communities in this fashion, and to have the consumers of the mathematical technology (so to speak) talk to the producers is the most productive way to spur further advancements in this area.